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4,201 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1345 J, W = 83.99 J | A thermodynamic system exchanges heat Q = 1345 J with its surroundings and performs work W = 83.99 J. By the first law, the change in internal energy is ΔU = Q − W = 1261 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,202 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -369.6 J, W = 425.1 J | A thermodynamic system exchanges heat Q = -369.6 J with its surroundings and performs work W = 425.1 J. By the first law, the change in internal energy is ΔU = Q − W = -794.7 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,203 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1239 J, W = -112.5 J | A thermodynamic system exchanges heat Q = 1239 J with its surroundings and performs work W = -112.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1351 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,204 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 567.1 J, W = -579.6 J | A thermodynamic system exchanges heat Q = 567.1 J with its surroundings and performs work W = -579.6 J. By the first law, the change in internal energy is ΔU = Q − W = 1147 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,205 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 594.9 J, W = 635.4 J | A thermodynamic system exchanges heat Q = 594.9 J with its surroundings and performs work W = 635.4 J. By the first law, the change in internal energy is ΔU = Q − W = -40.55 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,206 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1956 J, W = -211.7 J | A thermodynamic system exchanges heat Q = 1956 J with its surroundings and performs work W = -211.7 J. By the first law, the change in internal energy is ΔU = Q − W = 2168 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,207 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -15.89 J, W = -101.6 J | A thermodynamic system exchanges heat Q = -15.89 J with its surroundings and performs work W = -101.6 J. By the first law, the change in internal energy is ΔU = Q − W = 85.67 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,208 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -355.2 J, W = -428.7 J | A thermodynamic system exchanges heat Q = -355.2 J with its surroundings and performs work W = -428.7 J. By the first law, the change in internal energy is ΔU = Q − W = 73.55 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,209 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 971.5 J, W = -436.6 J | A thermodynamic system exchanges heat Q = 971.5 J with its surroundings and performs work W = -436.6 J. By the first law, the change in internal energy is ΔU = Q − W = 1408 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,210 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1369 J, W = 621.3 J | A thermodynamic system exchanges heat Q = 1369 J with its surroundings and performs work W = 621.3 J. By the first law, the change in internal energy is ΔU = Q − W = 747.4 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,211 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1483 J, W = 108.9 J | A thermodynamic system exchanges heat Q = 1483 J with its surroundings and performs work W = 108.9 J. By the first law, the change in internal energy is ΔU = Q − W = 1374 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,212 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1977 J, W = -776.2 J | A thermodynamic system exchanges heat Q = 1977 J with its surroundings and performs work W = -776.2 J. By the first law, the change in internal energy is ΔU = Q − W = 2754 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,213 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1300 J, W = 500.2 J | A thermodynamic system exchanges heat Q = 1300 J with its surroundings and performs work W = 500.2 J. By the first law, the change in internal energy is ΔU = Q − W = 799.4 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,214 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -308.5 J, W = -597.3 J | A thermodynamic system exchanges heat Q = -308.5 J with its surroundings and performs work W = -597.3 J. By the first law, the change in internal energy is ΔU = Q − W = 288.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,215 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -48.52 J, W = -730 J | A thermodynamic system exchanges heat Q = -48.52 J with its surroundings and performs work W = -730 J. By the first law, the change in internal energy is ΔU = Q − W = 681.5 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,216 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 374.2 J, W = 248.3 J | A thermodynamic system exchanges heat Q = 374.2 J with its surroundings and performs work W = 248.3 J. By the first law, the change in internal energy is ΔU = Q − W = 125.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,217 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 110.6 J, W = -332.1 J | A thermodynamic system exchanges heat Q = 110.6 J with its surroundings and performs work W = -332.1 J. By the first law, the change in internal energy is ΔU = Q − W = 442.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,218 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1015 J, W = -34.08 J | A thermodynamic system exchanges heat Q = 1015 J with its surroundings and performs work W = -34.08 J. By the first law, the change in internal energy is ΔU = Q − W = 1049 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,219 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -194.1 J, W = -677.1 J | A thermodynamic system exchanges heat Q = -194.1 J with its surroundings and performs work W = -677.1 J. By the first law, the change in internal energy is ΔU = Q − W = 483.1 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,220 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 125.5 J, W = 69.83 J | A thermodynamic system exchanges heat Q = 125.5 J with its surroundings and performs work W = 69.83 J. By the first law, the change in internal energy is ΔU = Q − W = 55.66 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,221 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 737.3 J, W = 376.3 J | A thermodynamic system exchanges heat Q = 737.3 J with its surroundings and performs work W = 376.3 J. By the first law, the change in internal energy is ΔU = Q − W = 361 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,222 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 392.4 J, W = 664.2 J | A thermodynamic system exchanges heat Q = 392.4 J with its surroundings and performs work W = 664.2 J. By the first law, the change in internal energy is ΔU = Q − W = -271.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,223 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 50.72 J, W = 341 J | A thermodynamic system exchanges heat Q = 50.72 J with its surroundings and performs work W = 341 J. By the first law, the change in internal energy is ΔU = Q − W = -290.3 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,224 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1854 J, W = 448.3 J | A thermodynamic system exchanges heat Q = 1854 J with its surroundings and performs work W = 448.3 J. By the first law, the change in internal energy is ΔU = Q − W = 1405 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,225 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 231.7 J, W = 245.8 J | A thermodynamic system exchanges heat Q = 231.7 J with its surroundings and performs work W = 245.8 J. By the first law, the change in internal energy is ΔU = Q − W = -14.12 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,226 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 817.1 J, W = -371.9 J | A thermodynamic system exchanges heat Q = 817.1 J with its surroundings and performs work W = -371.9 J. By the first law, the change in internal energy is ΔU = Q − W = 1189 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,227 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1244 J, W = 5.513 J | A thermodynamic system exchanges heat Q = 1244 J with its surroundings and performs work W = 5.513 J. By the first law, the change in internal energy is ΔU = Q − W = 1239 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,228 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -365 J, W = 640.7 J | A thermodynamic system exchanges heat Q = -365 J with its surroundings and performs work W = 640.7 J. By the first law, the change in internal energy is ΔU = Q − W = -1006 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,229 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1264 J, W = 195 J | A thermodynamic system exchanges heat Q = 1264 J with its surroundings and performs work W = 195 J. By the first law, the change in internal energy is ΔU = Q − W = 1069 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,230 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 199.5 J, W = -474.3 J | A thermodynamic system exchanges heat Q = 199.5 J with its surroundings and performs work W = -474.3 J. By the first law, the change in internal energy is ΔU = Q − W = 673.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,231 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -157.2 J, W = 761.5 J | A thermodynamic system exchanges heat Q = -157.2 J with its surroundings and performs work W = 761.5 J. By the first law, the change in internal energy is ΔU = Q − W = -918.7 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,232 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 76.34 J, W = 387.1 J | A thermodynamic system exchanges heat Q = 76.34 J with its surroundings and performs work W = 387.1 J. By the first law, the change in internal energy is ΔU = Q − W = -310.7 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,233 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 26.92 J, W = -204.2 J | A thermodynamic system exchanges heat Q = 26.92 J with its surroundings and performs work W = -204.2 J. By the first law, the change in internal energy is ΔU = Q − W = 231.1 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,234 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1541 J, W = -572.9 J | A thermodynamic system exchanges heat Q = 1541 J with its surroundings and performs work W = -572.9 J. By the first law, the change in internal energy is ΔU = Q − W = 2114 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,235 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1945 J, W = 317.3 J | A thermodynamic system exchanges heat Q = 1945 J with its surroundings and performs work W = 317.3 J. By the first law, the change in internal energy is ΔU = Q − W = 1627 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,236 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1125 J, W = -73.98 J | A thermodynamic system exchanges heat Q = 1125 J with its surroundings and performs work W = -73.98 J. By the first law, the change in internal energy is ΔU = Q − W = 1199 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,237 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -214.4 J, W = -1.748 J | A thermodynamic system exchanges heat Q = -214.4 J with its surroundings and performs work W = -1.748 J. By the first law, the change in internal energy is ΔU = Q − W = -212.7 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,238 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -415.6 J, W = -355.3 J | A thermodynamic system exchanges heat Q = -415.6 J with its surroundings and performs work W = -355.3 J. By the first law, the change in internal energy is ΔU = Q − W = -60.36 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,239 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 460.4 J, W = 523.5 J | A thermodynamic system exchanges heat Q = 460.4 J with its surroundings and performs work W = 523.5 J. By the first law, the change in internal energy is ΔU = Q − W = -63.04 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,240 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -288.8 J, W = -353.5 J | A thermodynamic system exchanges heat Q = -288.8 J with its surroundings and performs work W = -353.5 J. By the first law, the change in internal energy is ΔU = Q − W = 64.69 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,241 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1024 J, W = -7.428 J | A thermodynamic system exchanges heat Q = 1024 J with its surroundings and performs work W = -7.428 J. By the first law, the change in internal energy is ΔU = Q − W = 1032 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,242 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1657 J, W = -184.5 J | A thermodynamic system exchanges heat Q = 1657 J with its surroundings and performs work W = -184.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1841 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,243 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -389 J, W = -470.8 J | A thermodynamic system exchanges heat Q = -389 J with its surroundings and performs work W = -470.8 J. By the first law, the change in internal energy is ΔU = Q − W = 81.87 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,244 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -38.93 J, W = -110.3 J | A thermodynamic system exchanges heat Q = -38.93 J with its surroundings and performs work W = -110.3 J. By the first law, the change in internal energy is ΔU = Q − W = 71.33 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,245 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -369.4 J, W = -311.8 J | A thermodynamic system exchanges heat Q = -369.4 J with its surroundings and performs work W = -311.8 J. By the first law, the change in internal energy is ΔU = Q − W = -57.59 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,246 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 853.9 J, W = -243.5 J | A thermodynamic system exchanges heat Q = 853.9 J with its surroundings and performs work W = -243.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1097 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,247 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -435 J, W = 174 J | A thermodynamic system exchanges heat Q = -435 J with its surroundings and performs work W = 174 J. By the first law, the change in internal energy is ΔU = Q − W = -609 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,248 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1733 J, W = 676.3 J | A thermodynamic system exchanges heat Q = 1733 J with its surroundings and performs work W = 676.3 J. By the first law, the change in internal energy is ΔU = Q − W = 1057 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,249 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1769 J, W = 496.4 J | A thermodynamic system exchanges heat Q = 1769 J with its surroundings and performs work W = 496.4 J. By the first law, the change in internal energy is ΔU = Q − W = 1272 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,250 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 977.4 J, W = 411.6 J | A thermodynamic system exchanges heat Q = 977.4 J with its surroundings and performs work W = 411.6 J. By the first law, the change in internal energy is ΔU = Q − W = 565.8 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,251 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 352.4 J, W = 526.8 J | A thermodynamic system exchanges heat Q = 352.4 J with its surroundings and performs work W = 526.8 J. By the first law, the change in internal energy is ΔU = Q − W = -174.4 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,252 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 585.2 J, W = -115.9 J | A thermodynamic system exchanges heat Q = 585.2 J with its surroundings and performs work W = -115.9 J. By the first law, the change in internal energy is ΔU = Q − W = 701.1 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,253 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1323 J, W = 376.7 J | A thermodynamic system exchanges heat Q = 1323 J with its surroundings and performs work W = 376.7 J. By the first law, the change in internal energy is ΔU = Q − W = 946.4 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,254 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -357.2 J, W = -625.3 J | A thermodynamic system exchanges heat Q = -357.2 J with its surroundings and performs work W = -625.3 J. By the first law, the change in internal energy is ΔU = Q − W = 268.1 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,255 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 470.3 J, W = 354.1 J | A thermodynamic system exchanges heat Q = 470.3 J with its surroundings and performs work W = 354.1 J. By the first law, the change in internal energy is ΔU = Q − W = 116.2 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,256 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1195 J, W = 573.6 J | A thermodynamic system exchanges heat Q = 1195 J with its surroundings and performs work W = 573.6 J. By the first law, the change in internal energy is ΔU = Q − W = 621.3 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,257 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1845 J, W = -58.4 J | A thermodynamic system exchanges heat Q = 1845 J with its surroundings and performs work W = -58.4 J. By the first law, the change in internal energy is ΔU = Q − W = 1903 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,258 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 931.5 J, W = 109.3 J | A thermodynamic system exchanges heat Q = 931.5 J with its surroundings and performs work W = 109.3 J. By the first law, the change in internal energy is ΔU = Q − W = 822.2 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,259 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 797.3 J, W = -539.9 J | A thermodynamic system exchanges heat Q = 797.3 J with its surroundings and performs work W = -539.9 J. By the first law, the change in internal energy is ΔU = Q − W = 1337 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
4,260 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4241 c | A clock moving at velocity v = 0.4241 c relative to an inertial observer measures a proper time interval Δτ = 7.456 s. The observer measures a dilated interval Δt = γ Δτ = 8.233 s, where γ = 1 / sqrt(1 − v²/c²) = 1.104. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,261 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.886 c | A clock moving at velocity v = 0.886 c relative to an inertial observer measures a proper time interval Δτ = 1.319 s. The observer measures a dilated interval Δt = γ Δτ = 2.844 s, where γ = 1 / sqrt(1 − v²/c²) = 2.156. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,262 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1129 c | A clock moving at velocity v = 0.1129 c relative to an inertial observer measures a proper time interval Δτ = 7.996 s. The observer measures a dilated interval Δt = γ Δτ = 8.047 s, where γ = 1 / sqrt(1 − v²/c²) = 1.006. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,263 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9083 c | A clock moving at velocity v = 0.9083 c relative to an inertial observer measures a proper time interval Δτ = 0.8554 s. The observer measures a dilated interval Δt = γ Δτ = 2.045 s, where γ = 1 / sqrt(1 − v²/c²) = 2.391. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,264 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3124 c | A clock moving at velocity v = 0.3124 c relative to an inertial observer measures a proper time interval Δτ = 5.428 s. The observer measures a dilated interval Δt = γ Δτ = 5.714 s, where γ = 1 / sqrt(1 − v²/c²) = 1.053. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,265 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3869 c | A clock moving at velocity v = 0.3869 c relative to an inertial observer measures a proper time interval Δτ = 3.946 s. The observer measures a dilated interval Δt = γ Δτ = 4.279 s, where γ = 1 / sqrt(1 − v²/c²) = 1.084. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,266 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6232 c | A clock moving at velocity v = 0.6232 c relative to an inertial observer measures a proper time interval Δτ = 3.617 s. The observer measures a dilated interval Δt = γ Δτ = 4.625 s, where γ = 1 / sqrt(1 − v²/c²) = 1.279. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,267 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6603 c | A clock moving at velocity v = 0.6603 c relative to an inertial observer measures a proper time interval Δτ = 7.824 s. The observer measures a dilated interval Δt = γ Δτ = 10.42 s, where γ = 1 / sqrt(1 − v²/c²) = 1.332. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,268 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7887 c | A clock moving at velocity v = 0.7887 c relative to an inertial observer measures a proper time interval Δτ = 6.08 s. The observer measures a dilated interval Δt = γ Δτ = 9.89 s, where γ = 1 / sqrt(1 − v²/c²) = 1.627. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,269 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2057 c | A clock moving at velocity v = 0.2057 c relative to an inertial observer measures a proper time interval Δτ = 5.684 s. The observer measures a dilated interval Δt = γ Δτ = 5.809 s, where γ = 1 / sqrt(1 − v²/c²) = 1.022. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,270 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3728 c | A clock moving at velocity v = 0.3728 c relative to an inertial observer measures a proper time interval Δτ = 2.645 s. The observer measures a dilated interval Δt = γ Δτ = 2.851 s, where γ = 1 / sqrt(1 − v²/c²) = 1.078. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,271 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.252 c | A clock moving at velocity v = 0.252 c relative to an inertial observer measures a proper time interval Δτ = 4.627 s. The observer measures a dilated interval Δt = γ Δτ = 4.782 s, where γ = 1 / sqrt(1 − v²/c²) = 1.033. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,272 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4739 c | A clock moving at velocity v = 0.4739 c relative to an inertial observer measures a proper time interval Δτ = 4.664 s. The observer measures a dilated interval Δt = γ Δτ = 5.297 s, where γ = 1 / sqrt(1 − v²/c²) = 1.136. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,273 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.921 c | A clock moving at velocity v = 0.921 c relative to an inertial observer measures a proper time interval Δτ = 4.802 s. The observer measures a dilated interval Δt = γ Δτ = 12.33 s, where γ = 1 / sqrt(1 − v²/c²) = 2.567. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,274 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8177 c | A clock moving at velocity v = 0.8177 c relative to an inertial observer measures a proper time interval Δτ = 2.178 s. The observer measures a dilated interval Δt = γ Δτ = 3.784 s, where γ = 1 / sqrt(1 − v²/c²) = 1.737. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,275 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1498 c | A clock moving at velocity v = 0.1498 c relative to an inertial observer measures a proper time interval Δτ = 9.388 s. The observer measures a dilated interval Δt = γ Δτ = 9.495 s, where γ = 1 / sqrt(1 − v²/c²) = 1.011. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,276 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.335 c | A clock moving at velocity v = 0.335 c relative to an inertial observer measures a proper time interval Δτ = 6.103 s. The observer measures a dilated interval Δt = γ Δτ = 6.478 s, where γ = 1 / sqrt(1 − v²/c²) = 1.061. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,277 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3669 c | A clock moving at velocity v = 0.3669 c relative to an inertial observer measures a proper time interval Δτ = 0.6404 s. The observer measures a dilated interval Δt = γ Δτ = 0.6885 s, where γ = 1 / sqrt(1 − v²/c²) = 1.075. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,278 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2528 c | A clock moving at velocity v = 0.2528 c relative to an inertial observer measures a proper time interval Δτ = 7.66 s. The observer measures a dilated interval Δt = γ Δτ = 7.917 s, where γ = 1 / sqrt(1 − v²/c²) = 1.034. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,279 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3347 c | A clock moving at velocity v = 0.3347 c relative to an inertial observer measures a proper time interval Δτ = 8.691 s. The observer measures a dilated interval Δt = γ Δτ = 9.223 s, where γ = 1 / sqrt(1 − v²/c²) = 1.061. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,280 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2081 c | A clock moving at velocity v = 0.2081 c relative to an inertial observer measures a proper time interval Δτ = 6.303 s. The observer measures a dilated interval Δt = γ Δτ = 6.445 s, where γ = 1 / sqrt(1 − v²/c²) = 1.022. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,281 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2249 c | A clock moving at velocity v = 0.2249 c relative to an inertial observer measures a proper time interval Δτ = 6.179 s. The observer measures a dilated interval Δt = γ Δτ = 6.342 s, where γ = 1 / sqrt(1 − v²/c²) = 1.026. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,282 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1421 c | A clock moving at velocity v = 0.1421 c relative to an inertial observer measures a proper time interval Δτ = 1.144 s. The observer measures a dilated interval Δt = γ Δτ = 1.156 s, where γ = 1 / sqrt(1 − v²/c²) = 1.01. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,283 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1327 c | A clock moving at velocity v = 0.1327 c relative to an inertial observer measures a proper time interval Δτ = 6.844 s. The observer measures a dilated interval Δt = γ Δτ = 6.906 s, where γ = 1 / sqrt(1 − v²/c²) = 1.009. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,284 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8315 c | A clock moving at velocity v = 0.8315 c relative to an inertial observer measures a proper time interval Δτ = 5.799 s. The observer measures a dilated interval Δt = γ Δτ = 10.44 s, where γ = 1 / sqrt(1 − v²/c²) = 1.8. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,285 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1357 c | A clock moving at velocity v = 0.1357 c relative to an inertial observer measures a proper time interval Δτ = 7.174 s. The observer measures a dilated interval Δt = γ Δτ = 7.241 s, where γ = 1 / sqrt(1 − v²/c²) = 1.009. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,286 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1517 c | A clock moving at velocity v = 0.1517 c relative to an inertial observer measures a proper time interval Δτ = 7.145 s. The observer measures a dilated interval Δt = γ Δτ = 7.229 s, where γ = 1 / sqrt(1 − v²/c²) = 1.012. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,287 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3202 c | A clock moving at velocity v = 0.3202 c relative to an inertial observer measures a proper time interval Δτ = 2.574 s. The observer measures a dilated interval Δt = γ Δτ = 2.718 s, where γ = 1 / sqrt(1 − v²/c²) = 1.056. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,288 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6766 c | A clock moving at velocity v = 0.6766 c relative to an inertial observer measures a proper time interval Δτ = 8.575 s. The observer measures a dilated interval Δt = γ Δτ = 11.65 s, where γ = 1 / sqrt(1 − v²/c²) = 1.358. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,289 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4003 c | A clock moving at velocity v = 0.4003 c relative to an inertial observer measures a proper time interval Δτ = 7.183 s. The observer measures a dilated interval Δt = γ Δτ = 7.839 s, where γ = 1 / sqrt(1 − v²/c²) = 1.091. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,290 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6105 c | A clock moving at velocity v = 0.6105 c relative to an inertial observer measures a proper time interval Δτ = 3.497 s. The observer measures a dilated interval Δt = γ Δτ = 4.416 s, where γ = 1 / sqrt(1 − v²/c²) = 1.263. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,291 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1659 c | A clock moving at velocity v = 0.1659 c relative to an inertial observer measures a proper time interval Δτ = 4.438 s. The observer measures a dilated interval Δt = γ Δτ = 4.5 s, where γ = 1 / sqrt(1 − v²/c²) = 1.014. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,292 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5344 c | A clock moving at velocity v = 0.5344 c relative to an inertial observer measures a proper time interval Δτ = 2.37 s. The observer measures a dilated interval Δt = γ Δτ = 2.804 s, where γ = 1 / sqrt(1 − v²/c²) = 1.183. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,293 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3769 c | A clock moving at velocity v = 0.3769 c relative to an inertial observer measures a proper time interval Δτ = 3.714 s. The observer measures a dilated interval Δt = γ Δτ = 4.009 s, where γ = 1 / sqrt(1 − v²/c²) = 1.08. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,294 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7993 c | A clock moving at velocity v = 0.7993 c relative to an inertial observer measures a proper time interval Δτ = 4.16 s. The observer measures a dilated interval Δt = γ Δτ = 6.922 s, where γ = 1 / sqrt(1 − v²/c²) = 1.664. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,295 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6456 c | A clock moving at velocity v = 0.6456 c relative to an inertial observer measures a proper time interval Δτ = 8.759 s. The observer measures a dilated interval Δt = γ Δτ = 11.47 s, where γ = 1 / sqrt(1 − v²/c²) = 1.309. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,296 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2216 c | A clock moving at velocity v = 0.2216 c relative to an inertial observer measures a proper time interval Δτ = 3.265 s. The observer measures a dilated interval Δt = γ Δτ = 3.349 s, where γ = 1 / sqrt(1 − v²/c²) = 1.026. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,297 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7699 c | A clock moving at velocity v = 0.7699 c relative to an inertial observer measures a proper time interval Δτ = 3.35 s. The observer measures a dilated interval Δt = γ Δτ = 5.25 s, where γ = 1 / sqrt(1 − v²/c²) = 1.567. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,298 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4469 c | A clock moving at velocity v = 0.4469 c relative to an inertial observer measures a proper time interval Δτ = 6.52 s. The observer measures a dilated interval Δt = γ Δτ = 7.288 s, where γ = 1 / sqrt(1 − v²/c²) = 1.118. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,299 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5001 c | A clock moving at velocity v = 0.5001 c relative to an inertial observer measures a proper time interval Δτ = 9.735 s. The observer measures a dilated interval Δt = γ Δτ = 11.24 s, where γ = 1 / sqrt(1 − v²/c²) = 1.155. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
4,300 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.2192e-26 kg, speed 1.6040e+06 m/s | A free particle of mass 6.2192e-26 kg moving at speed 1.6040e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 6.6424e-15 m, where h is Planck's constant. This relation underlies the wave-particle duality of matter and is confirmed by electron diffraction experiments. | λ = h / p | wave_speed; classical momentum | Compute the de Broglie wavelength of a massive particle. |
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