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6,901 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['69.76', '18.32', '27.9', '34.95', '88.11', '56.02', '15.1', '84.09', '92.05', '61.89'], the sample mean is x̄ = 54.82 and the sample standard deviation is s = 29.16. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,902 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['17.44', '71.81', '54.99', '64.78', '20.37', '45.8', '67', '75.86', '20.4', '24.17', '88.01', '93.16'], the sample mean is x̄ = 53.65 and the sample standard deviation is s = 27.51. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,903 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['46.91', '77.74', '22.57', '20.36', '79.42', '70.61', '20.43', '44.43', '19.41', '92.2', '13.77', '65.58'], the sample mean is x̄ = 47.79 and the sample standard deviation is s = 28.31. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,904 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['50.18', '75.07', '69.55', '93.3', '78.4'], the sample mean is x̄ = 73.3 and the sample standard deviation is s = 15.63. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,905 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['27.82', '66.07', '75.67', '13.33', '61.47', '48.83', '40.36', '58.25'], the sample mean is x̄ = 48.97 and the sample standard deviation is s = 20.84. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty esti... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,906 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['50.98', '88.92', '83.01', '79.9', '19.33', '78.79', '35.38'], the sample mean is x̄ = 62.33 and the sample standard deviation is s = 27.14. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in m... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,907 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 6 observations | Given the sample values ['18.11', '12.04', '60.25', '45.53', '57.23', '92.97'], the sample mean is x̄ = 47.69 and the sample standard deviation is s = 29.83. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measureme... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,908 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['61.88', '39.4', '22.12', '63.67', '43.8'], the sample mean is x̄ = 46.17 and the sample standard deviation is s = 17.2. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,909 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['85.6', '63.23', '73.06', '43.01', '97.09'], the sample mean is x̄ = 72.4 and the sample standard deviation is s = 20.81. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,910 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['85.57', '15.94', '76.79', '68.96', '82.48', '29.32', '70.03', '63.66', '21.54', '72.51', '54.69'], the sample mean is x̄ = 58.32 and the sample standard deviation is s = 24.8. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,911 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['79.87', '30.6', '50.84', '22.77', '27.17'], the sample mean is x̄ = 42.25 and the sample standard deviation is s = 23.62. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,912 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['59.47', '58.57', '44.86', '78.63', '21.73', '86.17', '33.34', '16.75', '76.84', '24.06'], the sample mean is x̄ = 50.04 and the sample standard deviation is s = 25.59. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis fo... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,913 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['99.11', '93.72', '48.02', '42.56', '24.58', '30.77', '54.54'], the sample mean is x̄ = 56.19 and the sample standard deviation is s = 29.3. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in m... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,914 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['24.97', '65.75', '49.88', '74.94', '17.68'], the sample mean is x̄ = 46.64 and the sample standard deviation is s = 24.92. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,915 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['87.58', '12.01', '30.54', '21.38', '35.33', '68.51', '91.1', '40.82', '33.5', '87.76', '84.5'], the sample mean is x̄ = 53.91 and the sample standard deviation is s = 30.17. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the ba... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,916 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['29.97', '73.93', '35.69', '32.47', '21.22'], the sample mean is x̄ = 38.66 and the sample standard deviation is s = 20.44. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,917 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['91.72', '46.5', '54.67', '92.28', '67.59', '43.17', '90.37', '37.74', '15.84', '49.85'], the sample mean is x̄ = 58.97 and the sample standard deviation is s = 25.95. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,918 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['43.48', '48.27', '25.52', '32.15', '53.16', '19.58', '85.31', '32.6', '49.08', '19.95'], the sample mean is x̄ = 40.91 and the sample standard deviation is s = 19.79. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,919 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['79.22', '81.75', '56.28', '13.49', '12.4'], the sample mean is x̄ = 48.63 and the sample standard deviation is s = 34.06. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,920 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['59.85', '19.03', '45.88', '78.71', '18.1', '24.57', '56.47', '58.99', '56.52', '83.11', '70.32'], the sample mean is x̄ = 51.96 and the sample standard deviation is s = 22.79. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,921 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['19.67', '63.35', '88.16', '19.74', '40.84', '15.47', '17.6', '55.75', '46.77', '19.15', '14.85', '74.14'], the sample mean is x̄ = 39.62 and the sample standard deviation is s = 25.79. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,922 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['17.17', '14.3', '88.32', '42.5', '57.76', '34.8', '37.02', '82.09', '25.75', '73.37'], the sample mean is x̄ = 47.31 and the sample standard deviation is s = 26.72. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for u... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,923 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['11.57', '49.16', '41.4', '87.35', '70.87'], the sample mean is x̄ = 52.07 and the sample standard deviation is s = 28.99. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,924 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['48.33', '48.65', '32.84', '12.67', '20.84', '10.49', '27.91', '70.39', '88.5', '28.03', '46.23', '84.03'], the sample mean is x̄ = 43.24 and the sample standard deviation is s = 26.28. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,925 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['92.98', '85.45', '90.07', '49.31', '23.4', '41.79', '15.77', '31.65', '22.44', '86.84', '11.71', '15.18'], the sample mean is x̄ = 47.22 and the sample standard deviation is s = 32.61. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,926 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['46.27', '48.3', '37.95', '39.73', '63.04', '77.96', '81.56', '14.55', '62.97'], the sample mean is x̄ = 52.48 and the sample standard deviation is s = 21.17. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertai... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,927 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['21.19', '32.71', '11.77', '68.96', '62.7', '58.95', '94.73', '16.12', '74.34', '78.16', '87.21', '66.8'], the sample mean is x̄ = 56.14 and the sample standard deviation is s = 28.51. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,928 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['61.68', '38.35', '70.09', '73.7', '38.5', '25.49', '33', '64.8', '13.41'], the sample mean is x̄ = 46.56 and the sample standard deviation is s = 21.54. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty e... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,929 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['68.19', '63.81', '54.72', '92.13', '80.5', '57.63', '54.35', '30.13', '15.86'], the sample mean is x̄ = 57.48 and the sample standard deviation is s = 23.41. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertai... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,930 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['63.13', '71.66', '83.23', '44.4', '60.09', '36.34', '46.98', '44.38', '18.96', '29.47', '35.8', '14.42'], the sample mean is x̄ = 45.74 and the sample standard deviation is s = 20.76. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,931 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['61.39', '32.34', '84.78', '78.6', '49.32', '28.98', '72', '56.38', '63.19', '36.02', '61.16', '20.25'], the sample mean is x̄ = 53.7 and the sample standard deviation is s = 20.56. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,932 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['82.63', '16.03', '84.03', '33.77', '89.51', '86.33', '96.02', '22.56', '29.29', '14.59', '51.47', '91.37'], the sample mean is x̄ = 58.13 and the sample standard deviation is s = 33.02. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean a... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,933 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['19.51', '46.97', '85.69', '69.47', '69.33', '66.77', '69.44'], the sample mean is x̄ = 61.03 and the sample standard deviation is s = 21.5. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in m... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,934 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['18.56', '93.78', '94.29', '94.93', '81.93', '31.17', '71.59', '77.58', '61.44'], the sample mean is x̄ = 69.47 and the sample standard deviation is s = 27.85. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncerta... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,935 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['91.07', '90.77', '17.52', '50', '12.02', '85.27', '55.33', '62.89', '36.08', '69.53', '12.75', '83.83'], the sample mean is x̄ = 55.59 and the sample standard deviation is s = 30.18. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,936 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 6 observations | Given the sample values ['97.35', '93.53', '90.47', '37.37', '22.44', '78.2'], the sample mean is x̄ = 69.89 and the sample standard deviation is s = 31.98. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measuremen... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,937 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['95.21', '89.51', '56.98', '76.37', '30.98', '98.46', '97.53', '43.64', '93.93', '58.52', '15.55', '12.31'], the sample mean is x̄ = 64.08 and the sample standard deviation is s = 32.46. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean a... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,938 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['47.69', '38.82', '69.99', '84.59', '26.18'], the sample mean is x̄ = 53.46 and the sample standard deviation is s = 23.63. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,939 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['44.22', '17.66', '63.61', '18.24', '68.32', '17.56', '24.2', '90.19'], the sample mean is x̄ = 43 and the sample standard deviation is s = 28.16. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimate... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,940 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['81.23', '63.99', '48.11', '91.34', '23.54', '75.36', '45.29', '94.89', '70.32'], the sample mean is x̄ = 66.01 and the sample standard deviation is s = 23.38. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncerta... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,941 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['69.81', '14.5', '13.88', '41.92', '74.2', '33.28', '81.14', '81.11', '48.12', '41.91', '13.96', '90.96'], the sample mean is x̄ = 50.4 and the sample standard deviation is s = 28.4. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and i... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,942 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['25.93', '35.57', '90.44', '72.18', '60.24', '40', '43.96', '19.63'], the sample mean is x̄ = 48.49 and the sample standard deviation is s = 24.08. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimat... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,943 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['62.29', '87.31', '69.28', '67.71', '26.36', '67.6', '89.63', '93.36', '73.18', '35.47', '64.88', '44.54'], the sample mean is x̄ = 65.13 and the sample standard deviation is s = 20.89. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,944 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['74.83', '94.2', '48.64', '33.91', '56.05', '23.65', '49.96', '23.77'], the sample mean is x̄ = 50.63 and the sample standard deviation is s = 24.63. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estim... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,945 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['19.2', '22.81', '98.31', '42', '78.4', '84.3', '46.6', '58.84', '21.26', '55.23'], the sample mean is x̄ = 52.69 and the sample standard deviation is s = 27.77. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncer... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,946 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['33.93', '46.01', '72.7', '30.52', '13.7', '49.23', '98.65', '45.57'], the sample mean is x̄ = 48.79 and the sample standard deviation is s = 26.36. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estima... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,947 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['99.15', '77.76', '50.07', '61.82', '48.24', '66.09', '15.71'], the sample mean is x̄ = 59.83 and the sample standard deviation is s = 26.1. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in m... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,948 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['16.65', '99.88', '20.17', '90.65', '99.48', '32.35', '69.51', '98.84', '67.61'], the sample mean is x̄ = 66.12 and the sample standard deviation is s = 34.71. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncerta... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
6,949 | scientific_method | foundations | observation | 1 | explanation | The Role of Observation in Science | Science begins with careful, systematic observation of the natural world. Observations may be qualitative (descriptive) or quantitative (measured). Reliable observations are repeatable by independent observers under comparable conditions. Instruments extend human senses; calibration and uncertainty quantification are e... | null | null | Understand that science starts from reliable, recordable observation. |
6,950 | scientific_method | foundations | hypothesis | 2 | explanation | Formulating Testable Hypotheses | A scientific hypothesis is a proposed explanation for a set of observations. It must be falsifiable: there must exist conceivable evidence that would demonstrate the hypothesis to be incorrect. Hypotheses are stated so that they generate specific, testable predictions. Strong hypotheses are consistent with existing wel... | null | observation | Distinguish a scientific hypothesis from a conjecture and state its requirements. |
6,951 | scientific_method | foundations | experimentation | 3 | explanation | Controlled Experimentation and Variables | A controlled experiment isolates the effect of one or more independent variables on a dependent variable while holding confounding factors constant (control variables). Random assignment and blinding reduce bias. Replication increases statistical power and reveals variability. Experimental design must anticipate source... | null | hypothesis | Design a simple controlled experiment identifying independent, dependent, and control variables. |
6,952 | scientific_method | foundations | theory_and_law | 4 | explanation | Scientific Theories and Laws | A scientific law is a concise, often mathematical, description of a regular relationship observed in nature (e.g., conservation of energy, Newton's law of universal gravitation). A scientific theory is a coherent, well-substantiated explanatory framework that accounts for a broad range of observations and laws (e.g., t... | null | experimentation | Differentiate scientific laws from theories and explain their complementary roles. |
6,953 | scientific_method | foundations | peer_review_and_reproducibility | 5 | explanation | Peer Review, Reproducibility, and the Self-Correcting Nature of Science | Scientific claims gain credibility through independent scrutiny. Peer review evaluates methodology, analysis, and interpretation before formal publication. Reproducibility requires that independent researchers, following the same methods with equivalent materials, obtain statistically consistent results. Failures of re... | null | theory_and_law | Explain why reproducibility and peer review are essential to scientific reliability. |
6,954 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=4.897 m/s, a=7.311 m/s²) | An object starts with initial velocity 4.897 m/s and experiences constant acceleration 7.311 m/s² for 4.887 s. Final velocity: v = v0 + a t = 4.897 + (7.311)(4.887) = 40.63 m/s. Displacement: s = v0 t + (1/2) a t² = 111.2 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,955 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=12.22 m/s, a=-1.51 m/s²) | An object starts with initial velocity 12.22 m/s and experiences constant acceleration -1.51 m/s² for 10.03 s. Final velocity: v = v0 + a t = 12.22 + (-1.51)(10.03) = -2.922 m/s. Displacement: s = v0 t + (1/2) a t² = 46.62 m. These relations follow directly from the definitions of average velocity and constant accelera... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,956 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=12.23 m/s, a=-4.994 m/s²) | An object starts with initial velocity 12.23 m/s and experiences constant acceleration -4.994 m/s² for 4.545 s. Final velocity: v = v0 + a t = 12.23 + (-4.994)(4.545) = -10.47 m/s. Displacement: s = v0 t + (1/2) a t² = 4.004 m. These relations follow directly from the definitions of average velocity and constant accele... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,957 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=7.459 m/s, a=8.775 m/s²) | An object starts with initial velocity 7.459 m/s and experiences constant acceleration 8.775 m/s² for 7.103 s. Final velocity: v = v0 + a t = 7.459 + (8.775)(7.103) = 69.79 m/s. Displacement: s = v0 t + (1/2) a t² = 274.4 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,958 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=15.33 m/s, a=1.009 m/s²) | An object starts with initial velocity 15.33 m/s and experiences constant acceleration 1.009 m/s² for 15.82 s. Final velocity: v = v0 + a t = 15.33 + (1.009)(15.82) = 31.29 m/s. Displacement: s = v0 t + (1/2) a t² = 368.7 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,959 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=21.01 m/s, a=8.65 m/s²) | An object starts with initial velocity 21.01 m/s and experiences constant acceleration 8.65 m/s² for 7.846 s. Final velocity: v = v0 + a t = 21.01 + (8.65)(7.846) = 88.88 m/s. Displacement: s = v0 t + (1/2) a t² = 431.1 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,960 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=28.68 m/s, a=6.257 m/s²) | An object starts with initial velocity 28.68 m/s and experiences constant acceleration 6.257 m/s² for 10.42 s. Final velocity: v = v0 + a t = 28.68 + (6.257)(10.42) = 93.86 m/s. Displacement: s = v0 t + (1/2) a t² = 638.2 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,961 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=7.865 m/s, a=4.244 m/s²) | An object starts with initial velocity 7.865 m/s and experiences constant acceleration 4.244 m/s² for 9.149 s. Final velocity: v = v0 + a t = 7.865 + (4.244)(9.149) = 46.7 m/s. Displacement: s = v0 t + (1/2) a t² = 249.6 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,962 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=19.51 m/s, a=8.096 m/s²) | An object starts with initial velocity 19.51 m/s and experiences constant acceleration 8.096 m/s² for 14.24 s. Final velocity: v = v0 + a t = 19.51 + (8.096)(14.24) = 134.8 m/s. Displacement: s = v0 t + (1/2) a t² = 1099 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,963 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=8.208 m/s, a=6.815 m/s²) | An object starts with initial velocity 8.208 m/s and experiences constant acceleration 6.815 m/s² for 8.911 s. Final velocity: v = v0 + a t = 8.208 + (6.815)(8.911) = 68.94 m/s. Displacement: s = v0 t + (1/2) a t² = 343.8 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,964 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=25.35 m/s, a=2.051 m/s²) | An object starts with initial velocity 25.35 m/s and experiences constant acceleration 2.051 m/s² for 15.07 s. Final velocity: v = v0 + a t = 25.35 + (2.051)(15.07) = 56.25 m/s. Displacement: s = v0 t + (1/2) a t² = 614.7 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,965 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=26.9 m/s, a=8.97 m/s²) | An object starts with initial velocity 26.9 m/s and experiences constant acceleration 8.97 m/s² for 2.241 s. Final velocity: v = v0 + a t = 26.9 + (8.97)(2.241) = 47 m/s. Displacement: s = v0 t + (1/2) a t² = 82.82 m. These relations follow directly from the definitions of average velocity and constant acceleration. | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,966 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=15.47 m/s, a=5.503 m/s²) | An object starts with initial velocity 15.47 m/s and experiences constant acceleration 5.503 m/s² for 12.3 s. Final velocity: v = v0 + a t = 15.47 + (5.503)(12.3) = 83.14 m/s. Displacement: s = v0 t + (1/2) a t² = 606.3 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,967 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=17.2 m/s, a=3.199 m/s²) | An object starts with initial velocity 17.2 m/s and experiences constant acceleration 3.199 m/s² for 1.795 s. Final velocity: v = v0 + a t = 17.2 + (3.199)(1.795) = 22.94 m/s. Displacement: s = v0 t + (1/2) a t² = 36.02 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,968 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=16.23 m/s, a=8.613 m/s²) | An object starts with initial velocity 16.23 m/s and experiences constant acceleration 8.613 m/s² for 1.054 s. Final velocity: v = v0 + a t = 16.23 + (8.613)(1.054) = 25.31 m/s. Displacement: s = v0 t + (1/2) a t² = 21.9 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,969 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=12.02 m/s, a=7.5 m/s²) | An object starts with initial velocity 12.02 m/s and experiences constant acceleration 7.5 m/s² for 3.764 s. Final velocity: v = v0 + a t = 12.02 + (7.5)(3.764) = 40.25 m/s. Displacement: s = v0 t + (1/2) a t² = 98.37 m. These relations follow directly from the definitions of average velocity and constant acceleration. | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,970 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=8.145 m/s, a=1.643 m/s²) | An object starts with initial velocity 8.145 m/s and experiences constant acceleration 1.643 m/s² for 8.86 s. Final velocity: v = v0 + a t = 8.145 + (1.643)(8.86) = 22.7 m/s. Displacement: s = v0 t + (1/2) a t² = 136.6 m. These relations follow directly from the definitions of average velocity and constant acceleration... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,971 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=25.43 m/s, a=-3.431 m/s²) | An object starts with initial velocity 25.43 m/s and experiences constant acceleration -3.431 m/s² for 5.303 s. Final velocity: v = v0 + a t = 25.43 + (-3.431)(5.303) = 7.238 m/s. Displacement: s = v0 t + (1/2) a t² = 86.63 m. These relations follow directly from the definitions of average velocity and constant acceler... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,972 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=9.793 m/s, a=-2.229 m/s²) | An object starts with initial velocity 9.793 m/s and experiences constant acceleration -2.229 m/s² for 8.496 s. Final velocity: v = v0 + a t = 9.793 + (-2.229)(8.496) = -9.147 m/s. Displacement: s = v0 t + (1/2) a t² = 2.746 m. These relations follow directly from the definitions of average velocity and constant accele... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,973 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=11.3 m/s, a=-1.025 m/s²) | An object starts with initial velocity 11.3 m/s and experiences constant acceleration -1.025 m/s² for 9.022 s. Final velocity: v = v0 + a t = 11.3 + (-1.025)(9.022) = 2.052 m/s. Displacement: s = v0 t + (1/2) a t² = 60.21 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,974 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=19.9 m/s, a=4.261 m/s²) | An object starts with initial velocity 19.9 m/s and experiences constant acceleration 4.261 m/s² for 18.39 s. Final velocity: v = v0 + a t = 19.9 + (4.261)(18.39) = 98.25 m/s. Displacement: s = v0 t + (1/2) a t² = 1086 m. These relations follow directly from the definitions of average velocity and constant acceleration... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,975 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=20.97 m/s, a=3.727 m/s²) | An object starts with initial velocity 20.97 m/s and experiences constant acceleration 3.727 m/s² for 14.23 s. Final velocity: v = v0 + a t = 20.97 + (3.727)(14.23) = 74.01 m/s. Displacement: s = v0 t + (1/2) a t² = 675.9 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,976 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=16.49 m/s, a=-0.3258 m/s²) | An object starts with initial velocity 16.49 m/s and experiences constant acceleration -0.3258 m/s² for 10.63 s. Final velocity: v = v0 + a t = 16.49 + (-0.3258)(10.63) = 13.02 m/s. Displacement: s = v0 t + (1/2) a t² = 156.8 m. These relations follow directly from the definitions of average velocity and constant accel... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,977 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=17.02 m/s, a=5.33 m/s²) | An object starts with initial velocity 17.02 m/s and experiences constant acceleration 5.33 m/s² for 17.47 s. Final velocity: v = v0 + a t = 17.02 + (5.33)(17.47) = 110.1 m/s. Displacement: s = v0 t + (1/2) a t² = 1110 m. These relations follow directly from the definitions of average velocity and constant acceleration... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,978 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=9.82 m/s, a=3.825 m/s²) | An object starts with initial velocity 9.82 m/s and experiences constant acceleration 3.825 m/s² for 19.6 s. Final velocity: v = v0 + a t = 9.82 + (3.825)(19.6) = 84.79 m/s. Displacement: s = v0 t + (1/2) a t² = 927.1 m. These relations follow directly from the definitions of average velocity and constant acceleration. | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,979 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=24.11 m/s, a=-4.26 m/s²) | An object starts with initial velocity 24.11 m/s and experiences constant acceleration -4.26 m/s² for 3.358 s. Final velocity: v = v0 + a t = 24.11 + (-4.26)(3.358) = 9.8 m/s. Displacement: s = v0 t + (1/2) a t² = 56.93 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,980 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=4.043 m/s, a=3.987 m/s²) | An object starts with initial velocity 4.043 m/s and experiences constant acceleration 3.987 m/s² for 4.431 s. Final velocity: v = v0 + a t = 4.043 + (3.987)(4.431) = 21.71 m/s. Displacement: s = v0 t + (1/2) a t² = 57.05 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,981 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=25.34 m/s, a=6.716 m/s²) | An object starts with initial velocity 25.34 m/s and experiences constant acceleration 6.716 m/s² for 19.9 s. Final velocity: v = v0 + a t = 25.34 + (6.716)(19.9) = 159 m/s. Displacement: s = v0 t + (1/2) a t² = 1834 m. These relations follow directly from the definitions of average velocity and constant acceleration. | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,982 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=7.392 m/s, a=8.931 m/s²) | An object starts with initial velocity 7.392 m/s and experiences constant acceleration 8.931 m/s² for 17.89 s. Final velocity: v = v0 + a t = 7.392 + (8.931)(17.89) = 167.2 m/s. Displacement: s = v0 t + (1/2) a t² = 1562 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,983 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=1.335 m/s, a=-3.299 m/s²) | An object starts with initial velocity 1.335 m/s and experiences constant acceleration -3.299 m/s² for 10.02 s. Final velocity: v = v0 + a t = 1.335 + (-3.299)(10.02) = -31.71 m/s. Displacement: s = v0 t + (1/2) a t² = -152.1 m. These relations follow directly from the definitions of average velocity and constant accel... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,984 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=12.67 m/s, a=5.923 m/s²) | An object starts with initial velocity 12.67 m/s and experiences constant acceleration 5.923 m/s² for 5.314 s. Final velocity: v = v0 + a t = 12.67 + (5.923)(5.314) = 44.15 m/s. Displacement: s = v0 t + (1/2) a t² = 151 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,985 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=17.88 m/s, a=6.291 m/s²) | An object starts with initial velocity 17.88 m/s and experiences constant acceleration 6.291 m/s² for 11.55 s. Final velocity: v = v0 + a t = 17.88 + (6.291)(11.55) = 90.53 m/s. Displacement: s = v0 t + (1/2) a t² = 626.1 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,986 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=11.85 m/s, a=1.319 m/s²) | An object starts with initial velocity 11.85 m/s and experiences constant acceleration 1.319 m/s² for 5.723 s. Final velocity: v = v0 + a t = 11.85 + (1.319)(5.723) = 19.39 m/s. Displacement: s = v0 t + (1/2) a t² = 89.38 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,987 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=29.28 m/s, a=4.714 m/s²) | An object starts with initial velocity 29.28 m/s and experiences constant acceleration 4.714 m/s² for 2.9 s. Final velocity: v = v0 + a t = 29.28 + (4.714)(2.9) = 42.95 m/s. Displacement: s = v0 t + (1/2) a t² = 104.7 m. These relations follow directly from the definitions of average velocity and constant acceleration. | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,988 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=17.38 m/s, a=-0.08519 m/s²) | An object starts with initial velocity 17.38 m/s and experiences constant acceleration -0.08519 m/s² for 12.39 s. Final velocity: v = v0 + a t = 17.38 + (-0.08519)(12.39) = 16.32 m/s. Displacement: s = v0 t + (1/2) a t² = 208.8 m. These relations follow directly from the definitions of average velocity and constant acc... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,989 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=11.37 m/s, a=-4.885 m/s²) | An object starts with initial velocity 11.37 m/s and experiences constant acceleration -4.885 m/s² for 1.471 s. Final velocity: v = v0 + a t = 11.37 + (-4.885)(1.471) = 4.185 m/s. Displacement: s = v0 t + (1/2) a t² = 11.44 m. These relations follow directly from the definitions of average velocity and constant acceler... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,990 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=19.68 m/s, a=3.586 m/s²) | An object starts with initial velocity 19.68 m/s and experiences constant acceleration 3.586 m/s² for 16.33 s. Final velocity: v = v0 + a t = 19.68 + (3.586)(16.33) = 78.25 m/s. Displacement: s = v0 t + (1/2) a t² = 799.8 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,991 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=8.961 m/s, a=9.569 m/s²) | An object starts with initial velocity 8.961 m/s and experiences constant acceleration 9.569 m/s² for 5.754 s. Final velocity: v = v0 + a t = 8.961 + (9.569)(5.754) = 64.02 m/s. Displacement: s = v0 t + (1/2) a t² = 210 m. These relations follow directly from the definitions of average velocity and constant acceleratio... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,992 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=1.409 m/s, a=8.808 m/s²) | An object starts with initial velocity 1.409 m/s and experiences constant acceleration 8.808 m/s² for 19.97 s. Final velocity: v = v0 + a t = 1.409 + (8.808)(19.97) = 177.3 m/s. Displacement: s = v0 t + (1/2) a t² = 1785 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,993 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=20.47 m/s, a=-0.8904 m/s²) | An object starts with initial velocity 20.47 m/s and experiences constant acceleration -0.8904 m/s² for 18.16 s. Final velocity: v = v0 + a t = 20.47 + (-0.8904)(18.16) = 4.304 m/s. Displacement: s = v0 t + (1/2) a t² = 224.9 m. These relations follow directly from the definitions of average velocity and constant accel... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,994 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=25.85 m/s, a=4.201 m/s²) | An object starts with initial velocity 25.85 m/s and experiences constant acceleration 4.201 m/s² for 9.195 s. Final velocity: v = v0 + a t = 25.85 + (4.201)(9.195) = 64.48 m/s. Displacement: s = v0 t + (1/2) a t² = 415.3 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,995 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=12.31 m/s, a=-0.09688 m/s²) | An object starts with initial velocity 12.31 m/s and experiences constant acceleration -0.09688 m/s² for 9.28 s. Final velocity: v = v0 + a t = 12.31 + (-0.09688)(9.28) = 11.41 m/s. Displacement: s = v0 t + (1/2) a t² = 110.1 m. These relations follow directly from the definitions of average velocity and constant accel... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,996 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=4.855 m/s, a=7.303 m/s²) | An object starts with initial velocity 4.855 m/s and experiences constant acceleration 7.303 m/s² for 18.7 s. Final velocity: v = v0 + a t = 4.855 + (7.303)(18.7) = 141.4 m/s. Displacement: s = v0 t + (1/2) a t² = 1367 m. These relations follow directly from the definitions of average velocity and constant acceleration... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,997 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=9.442 m/s, a=7.978 m/s²) | An object starts with initial velocity 9.442 m/s and experiences constant acceleration 7.978 m/s² for 10.31 s. Final velocity: v = v0 + a t = 9.442 + (7.978)(10.31) = 91.7 m/s. Displacement: s = v0 t + (1/2) a t² = 521.4 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,998 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=13.29 m/s, a=3.063 m/s²) | An object starts with initial velocity 13.29 m/s and experiences constant acceleration 3.063 m/s² for 1.554 s. Final velocity: v = v0 + a t = 13.29 + (3.063)(1.554) = 18.05 m/s. Displacement: s = v0 t + (1/2) a t² = 24.36 m. These relations follow directly from the definitions of average velocity and constant accelerat... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
6,999 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=13.52 m/s, a=9.295 m/s²) | An object starts with initial velocity 13.52 m/s and experiences constant acceleration 9.295 m/s² for 15.59 s. Final velocity: v = v0 + a t = 13.52 + (9.295)(15.59) = 158.4 m/s. Displacement: s = v0 t + (1/2) a t² = 1341 m. These relations follow directly from the definitions of average velocity and constant accelerati... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
7,000 | physics | mechanics | kinematics_1d | 2 | worked_example | One-dimensional motion with constant acceleration (v0=28.94 m/s, a=-2.116 m/s²) | An object starts with initial velocity 28.94 m/s and experiences constant acceleration -2.116 m/s² for 10.62 s. Final velocity: v = v0 + a t = 28.94 + (-2.116)(10.62) = 6.47 m/s. Displacement: s = v0 t + (1/2) a t² = 188.1 m. These relations follow directly from the definitions of average velocity and constant accelera... | v = v_0 + a t; s = v_0 t + (1/2) a t^2; v^2 = v_0^2 + 2 a s | definition of velocity and acceleration | Apply the three kinematic equations for constant acceleration in one dimension. |
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