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7,701 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -453.7 J, W = 3.525 J | A thermodynamic system exchanges heat Q = -453.7 J with its surroundings and performs work W = 3.525 J. By the first law, the change in internal energy is ΔU = Q − W = -457.2 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. |
7,702 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 796.4 J, W = 13.72 J | A thermodynamic system exchanges heat Q = 796.4 J with its surroundings and performs work W = 13.72 J. By the first law, the change in internal energy is ΔU = Q − W = 782.7 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. |
7,703 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 950.3 J, W = -271.1 J | A thermodynamic system exchanges heat Q = 950.3 J with its surroundings and performs work W = -271.1 J. By the first law, the change in internal energy is ΔU = Q − W = 1221 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. |
7,704 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 177.4 J, W = -766.7 J | A thermodynamic system exchanges heat Q = 177.4 J with its surroundings and performs work W = -766.7 J. By the first law, the change in internal energy is ΔU = Q − W = 944.1 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. |
7,705 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -84.47 J, W = 576.3 J | A thermodynamic system exchanges heat Q = -84.47 J with its surroundings and performs work W = 576.3 J. By the first law, the change in internal energy is ΔU = Q − W = -660.8 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. |
7,706 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1035 J, W = -196 J | A thermodynamic system exchanges heat Q = 1035 J with its surroundings and performs work W = -196 J. By the first law, the change in internal energy is ΔU = Q − W = 1231 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. |
7,707 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 768.7 J, W = -241 J | A thermodynamic system exchanges heat Q = 768.7 J with its surroundings and performs work W = -241 J. By the first law, the change in internal energy is ΔU = Q − W = 1010 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. |
7,708 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1030 J, W = 599.9 J | A thermodynamic system exchanges heat Q = 1030 J with its surroundings and performs work W = 599.9 J. By the first law, the change in internal energy is ΔU = Q − W = 430.3 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. |
7,709 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -26.33 J, W = 291 J | A thermodynamic system exchanges heat Q = -26.33 J with its surroundings and performs work W = 291 J. By the first law, the change in internal energy is ΔU = Q − W = -317.3 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. |
7,710 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 742.1 J, W = 198.3 J | A thermodynamic system exchanges heat Q = 742.1 J with its surroundings and performs work W = 198.3 J. By the first law, the change in internal energy is ΔU = Q − W = 543.8 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. |
7,711 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1759 J, W = -164.4 J | A thermodynamic system exchanges heat Q = 1759 J with its surroundings and performs work W = -164.4 J. By the first law, the change in internal energy is ΔU = Q − W = 1924 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. |
7,712 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1087 J, W = -465.3 J | A thermodynamic system exchanges heat Q = 1087 J with its surroundings and performs work W = -465.3 J. By the first law, the change in internal energy is ΔU = Q − W = 1552 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. |
7,713 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1869 J, W = 156.4 J | A thermodynamic system exchanges heat Q = 1869 J with its surroundings and performs work W = 156.4 J. By the first law, the change in internal energy is ΔU = Q − W = 1713 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. |
7,714 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 291.3 J, W = -426.9 J | A thermodynamic system exchanges heat Q = 291.3 J with its surroundings and performs work W = -426.9 J. By the first law, the change in internal energy is ΔU = Q − W = 718.3 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. |
7,715 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -37.53 J, W = 674.1 J | A thermodynamic system exchanges heat Q = -37.53 J with its surroundings and performs work W = 674.1 J. By the first law, the change in internal energy is ΔU = Q − W = -711.6 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. |
7,716 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 685.4 J, W = 471.7 J | A thermodynamic system exchanges heat Q = 685.4 J with its surroundings and performs work W = 471.7 J. By the first law, the change in internal energy is ΔU = Q − W = 213.7 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. |
7,717 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -99.96 J, W = -226.1 J | A thermodynamic system exchanges heat Q = -99.96 J with its surroundings and performs work W = -226.1 J. By the first law, the change in internal energy is ΔU = Q − W = 126.1 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. |
7,718 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -116 J, W = -130.6 J | A thermodynamic system exchanges heat Q = -116 J with its surroundings and performs work W = -130.6 J. By the first law, the change in internal energy is ΔU = Q − W = 14.65 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. |
7,719 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1715 J, W = -241.1 J | A thermodynamic system exchanges heat Q = 1715 J with its surroundings and performs work W = -241.1 J. By the first law, the change in internal energy is ΔU = Q − W = 1956 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. |
7,720 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -290.7 J, W = 495.2 J | A thermodynamic system exchanges heat Q = -290.7 J with its surroundings and performs work W = 495.2 J. By the first law, the change in internal energy is ΔU = Q − W = -785.9 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. |
7,721 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -165.4 J, W = 246.6 J | A thermodynamic system exchanges heat Q = -165.4 J with its surroundings and performs work W = 246.6 J. By the first law, the change in internal energy is ΔU = Q − W = -412 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. |
7,722 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 931.8 J, W = 683.9 J | A thermodynamic system exchanges heat Q = 931.8 J with its surroundings and performs work W = 683.9 J. By the first law, the change in internal energy is ΔU = Q − W = 247.8 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. |
7,723 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -122.7 J, W = -342.7 J | A thermodynamic system exchanges heat Q = -122.7 J with its surroundings and performs work W = -342.7 J. By the first law, the change in internal energy is ΔU = Q − W = 219.9 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. |
7,724 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -321.7 J, W = 678.5 J | A thermodynamic system exchanges heat Q = -321.7 J with its surroundings and performs work W = 678.5 J. By the first law, the change in internal energy is ΔU = Q − W = -1000 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. |
7,725 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1888 J, W = 429.8 J | A thermodynamic system exchanges heat Q = 1888 J with its surroundings and performs work W = 429.8 J. By the first law, the change in internal energy is ΔU = Q − W = 1458 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. |
7,726 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 45.92 J, W = -557.4 J | A thermodynamic system exchanges heat Q = 45.92 J with its surroundings and performs work W = -557.4 J. By the first law, the change in internal energy is ΔU = Q − W = 603.3 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. |
7,727 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 294.4 J, W = 411.9 J | A thermodynamic system exchanges heat Q = 294.4 J with its surroundings and performs work W = 411.9 J. By the first law, the change in internal energy is ΔU = Q − W = -117.6 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. |
7,728 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1253 J, W = -156.4 J | A thermodynamic system exchanges heat Q = 1253 J with its surroundings and performs work W = -156.4 J. By the first law, the change in internal energy is ΔU = Q − W = 1409 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. |
7,729 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 589.7 J, W = -58.76 J | A thermodynamic system exchanges heat Q = 589.7 J with its surroundings and performs work W = -58.76 J. By the first law, the change in internal energy is ΔU = Q − W = 648.5 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. |
7,730 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1301 J, W = -553.3 J | A thermodynamic system exchanges heat Q = 1301 J with its surroundings and performs work W = -553.3 J. By the first law, the change in internal energy is ΔU = Q − W = 1854 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. |
7,731 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 521.5 J, W = 522.7 J | A thermodynamic system exchanges heat Q = 521.5 J with its surroundings and performs work W = 522.7 J. By the first law, the change in internal energy is ΔU = Q − W = -1.155 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. |
7,732 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1307 J, W = 262.5 J | A thermodynamic system exchanges heat Q = 1307 J with its surroundings and performs work W = 262.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1044 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. |
7,733 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1885 J, W = -731.6 J | A thermodynamic system exchanges heat Q = 1885 J with its surroundings and performs work W = -731.6 J. By the first law, the change in internal energy is ΔU = Q − W = 2616 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. |
7,734 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4204 c | A clock moving at velocity v = 0.4204 c relative to an inertial observer measures a proper time interval Δτ = 3.352 s. The observer measures a dilated interval Δt = γ Δτ = 3.695 s, where γ = 1 / sqrt(1 − v²/c²) = 1.102. 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. |
7,735 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3203 c | A clock moving at velocity v = 0.3203 c relative to an inertial observer measures a proper time interval Δτ = 5.144 s. The observer measures a dilated interval Δt = γ Δτ = 5.43 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. |
7,736 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6494 c | A clock moving at velocity v = 0.6494 c relative to an inertial observer measures a proper time interval Δτ = 5.512 s. The observer measures a dilated interval Δt = γ Δτ = 7.248 s, where γ = 1 / sqrt(1 − v²/c²) = 1.315. 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. |
7,737 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1825 c | A clock moving at velocity v = 0.1825 c relative to an inertial observer measures a proper time interval Δτ = 2.048 s. The observer measures a dilated interval Δt = γ Δτ = 2.083 s, where γ = 1 / sqrt(1 − v²/c²) = 1.017. 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. |
7,738 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1909 c | A clock moving at velocity v = 0.1909 c relative to an inertial observer measures a proper time interval Δτ = 6.547 s. The observer measures a dilated interval Δt = γ Δτ = 6.67 s, where γ = 1 / sqrt(1 − v²/c²) = 1.019. 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. |
7,739 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7462 c | A clock moving at velocity v = 0.7462 c relative to an inertial observer measures a proper time interval Δτ = 0.4045 s. The observer measures a dilated interval Δt = γ Δτ = 0.6076 s, where γ = 1 / sqrt(1 − v²/c²) = 1.502. 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. |
7,740 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3317 c | A clock moving at velocity v = 0.3317 c relative to an inertial observer measures a proper time interval Δτ = 5.344 s. The observer measures a dilated interval Δt = γ Δτ = 5.665 s, where γ = 1 / sqrt(1 − v²/c²) = 1.06. 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. |
7,741 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8233 c | A clock moving at velocity v = 0.8233 c relative to an inertial observer measures a proper time interval Δτ = 7.044 s. The observer measures a dilated interval Δt = γ Δτ = 12.41 s, where γ = 1 / sqrt(1 − v²/c²) = 1.762. 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. |
7,742 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2335 c | A clock moving at velocity v = 0.2335 c relative to an inertial observer measures a proper time interval Δτ = 8.189 s. The observer measures a dilated interval Δt = γ Δτ = 8.422 s, where γ = 1 / sqrt(1 − v²/c²) = 1.028. 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. |
7,743 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1712 c | A clock moving at velocity v = 0.1712 c relative to an inertial observer measures a proper time interval Δτ = 8.505 s. The observer measures a dilated interval Δt = γ Δτ = 8.632 s, where γ = 1 / sqrt(1 − v²/c²) = 1.015. 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. |
7,744 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1802 c | A clock moving at velocity v = 0.1802 c relative to an inertial observer measures a proper time interval Δτ = 8.288 s. The observer measures a dilated interval Δt = γ Δτ = 8.426 s, where γ = 1 / sqrt(1 − v²/c²) = 1.017. 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. |
7,745 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7648 c | A clock moving at velocity v = 0.7648 c relative to an inertial observer measures a proper time interval Δτ = 3.721 s. The observer measures a dilated interval Δt = γ Δτ = 5.776 s, where γ = 1 / sqrt(1 − v²/c²) = 1.552. 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. |
7,746 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3517 c | A clock moving at velocity v = 0.3517 c relative to an inertial observer measures a proper time interval Δτ = 6.627 s. The observer measures a dilated interval Δt = γ Δτ = 7.08 s, where γ = 1 / sqrt(1 − v²/c²) = 1.068. 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. |
7,747 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8509 c | A clock moving at velocity v = 0.8509 c relative to an inertial observer measures a proper time interval Δτ = 7.112 s. The observer measures a dilated interval Δt = γ Δτ = 13.54 s, where γ = 1 / sqrt(1 − v²/c²) = 1.904. 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. |
7,748 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5576 c | A clock moving at velocity v = 0.5576 c relative to an inertial observer measures a proper time interval Δτ = 8.697 s. The observer measures a dilated interval Δt = γ Δτ = 10.48 s, where γ = 1 / sqrt(1 − v²/c²) = 1.205. 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. |
7,749 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5619 c | A clock moving at velocity v = 0.5619 c relative to an inertial observer measures a proper time interval Δτ = 5.358 s. The observer measures a dilated interval Δt = γ Δτ = 6.477 s, where γ = 1 / sqrt(1 − v²/c²) = 1.209. 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. |
7,750 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9427 c | A clock moving at velocity v = 0.9427 c relative to an inertial observer measures a proper time interval Δτ = 2.98 s. The observer measures a dilated interval Δt = γ Δτ = 8.932 s, where γ = 1 / sqrt(1 − v²/c²) = 2.998. 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. |
7,751 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1072 c | A clock moving at velocity v = 0.1072 c relative to an inertial observer measures a proper time interval Δτ = 7.636 s. The observer measures a dilated interval Δt = γ Δτ = 7.68 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. |
7,752 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1933 c | A clock moving at velocity v = 0.1933 c relative to an inertial observer measures a proper time interval Δτ = 3.267 s. The observer measures a dilated interval Δt = γ Δτ = 3.33 s, where γ = 1 / sqrt(1 − v²/c²) = 1.019. 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. |
7,753 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5728 c | A clock moving at velocity v = 0.5728 c relative to an inertial observer measures a proper time interval Δτ = 0.5849 s. The observer measures a dilated interval Δt = γ Δτ = 0.7135 s, where γ = 1 / sqrt(1 − v²/c²) = 1.22. 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. |
7,754 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3629 c | A clock moving at velocity v = 0.3629 c relative to an inertial observer measures a proper time interval Δτ = 3.056 s. The observer measures a dilated interval Δt = γ Δτ = 3.28 s, where γ = 1 / sqrt(1 − v²/c²) = 1.073. 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. |
7,755 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2135 c | A clock moving at velocity v = 0.2135 c relative to an inertial observer measures a proper time interval Δτ = 8.368 s. The observer measures a dilated interval Δt = γ Δτ = 8.566 s, where γ = 1 / sqrt(1 − v²/c²) = 1.024. 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. |
7,756 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1008 c | A clock moving at velocity v = 0.1008 c relative to an inertial observer measures a proper time interval Δτ = 7.946 s. The observer measures a dilated interval Δt = γ Δτ = 7.987 s, where γ = 1 / sqrt(1 − v²/c²) = 1.005. 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. |
7,757 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.251 c | A clock moving at velocity v = 0.251 c relative to an inertial observer measures a proper time interval Δτ = 9.281 s. The observer measures a dilated interval Δt = γ Δτ = 9.588 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. |
7,758 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8782 c | A clock moving at velocity v = 0.8782 c relative to an inertial observer measures a proper time interval Δτ = 4.978 s. The observer measures a dilated interval Δt = γ Δτ = 10.41 s, where γ = 1 / sqrt(1 − v²/c²) = 2.091. 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. |
7,759 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6425 c | A clock moving at velocity v = 0.6425 c relative to an inertial observer measures a proper time interval Δτ = 4.804 s. The observer measures a dilated interval Δt = γ Δτ = 6.27 s, where γ = 1 / sqrt(1 − v²/c²) = 1.305. 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. |
7,760 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8071 c | A clock moving at velocity v = 0.8071 c relative to an inertial observer measures a proper time interval Δτ = 5.208 s. The observer measures a dilated interval Δt = γ Δτ = 8.821 s, where γ = 1 / sqrt(1 − v²/c²) = 1.694. 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. |
7,761 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7595 c | A clock moving at velocity v = 0.7595 c relative to an inertial observer measures a proper time interval Δτ = 9.598 s. The observer measures a dilated interval Δt = γ Δτ = 14.76 s, where γ = 1 / sqrt(1 − v²/c²) = 1.537. 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. |
7,762 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6421 c | A clock moving at velocity v = 0.6421 c relative to an inertial observer measures a proper time interval Δτ = 2.818 s. The observer measures a dilated interval Δt = γ Δτ = 3.676 s, where γ = 1 / sqrt(1 − v²/c²) = 1.304. 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. |
7,763 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8397 c | A clock moving at velocity v = 0.8397 c relative to an inertial observer measures a proper time interval Δτ = 7.489 s. The observer measures a dilated interval Δt = γ Δτ = 13.79 s, where γ = 1 / sqrt(1 − v²/c²) = 1.841. 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. |
7,764 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7588 c | A clock moving at velocity v = 0.7588 c relative to an inertial observer measures a proper time interval Δτ = 1.188 s. The observer measures a dilated interval Δt = γ Δτ = 1.823 s, where γ = 1 / sqrt(1 − v²/c²) = 1.535. 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. |
7,765 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.47 c | A clock moving at velocity v = 0.47 c relative to an inertial observer measures a proper time interval Δτ = 1.293 s. The observer measures a dilated interval Δt = γ Δτ = 1.465 s, where γ = 1 / sqrt(1 − v²/c²) = 1.133. 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. |
7,766 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3518 c | A clock moving at velocity v = 0.3518 c relative to an inertial observer measures a proper time interval Δτ = 1.803 s. The observer measures a dilated interval Δt = γ Δτ = 1.926 s, where γ = 1 / sqrt(1 − v²/c²) = 1.068. 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. |
7,767 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6189 c | A clock moving at velocity v = 0.6189 c relative to an inertial observer measures a proper time interval Δτ = 7.149 s. The observer measures a dilated interval Δt = γ Δτ = 9.101 s, where γ = 1 / sqrt(1 − v²/c²) = 1.273. 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. |
7,768 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4707 c | A clock moving at velocity v = 0.4707 c relative to an inertial observer measures a proper time interval Δτ = 1.448 s. The observer measures a dilated interval Δt = γ Δτ = 1.642 s, where γ = 1 / sqrt(1 − v²/c²) = 1.133. 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. |
7,769 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1638 c | A clock moving at velocity v = 0.1638 c relative to an inertial observer measures a proper time interval Δτ = 1.83 s. The observer measures a dilated interval Δt = γ Δτ = 1.855 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. |
7,770 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3598 c | A clock moving at velocity v = 0.3598 c relative to an inertial observer measures a proper time interval Δτ = 0.6396 s. The observer measures a dilated interval Δt = γ Δτ = 0.6855 s, where γ = 1 / sqrt(1 − v²/c²) = 1.072. 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. |
7,771 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3417 c | A clock moving at velocity v = 0.3417 c relative to an inertial observer measures a proper time interval Δτ = 6.117 s. The observer measures a dilated interval Δt = γ Δτ = 6.509 s, where γ = 1 / sqrt(1 − v²/c²) = 1.064. 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. |
7,772 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5521 c | A clock moving at velocity v = 0.5521 c relative to an inertial observer measures a proper time interval Δτ = 7.958 s. The observer measures a dilated interval Δt = γ Δτ = 9.545 s, where γ = 1 / sqrt(1 − v²/c²) = 1.199. 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. |
7,773 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1959 c | A clock moving at velocity v = 0.1959 c relative to an inertial observer measures a proper time interval Δτ = 7.327 s. The observer measures a dilated interval Δt = γ Δτ = 7.472 s, where γ = 1 / sqrt(1 − v²/c²) = 1.02. 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. |
7,774 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.3766e-26 kg, speed 9.1704e+06 m/s | A free particle of mass 9.3766e-26 kg moving at speed 9.1704e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.7058e-16 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. |
7,775 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.6836e-26 kg, speed 8.6189e+06 m/s | A free particle of mass 5.6836e-26 kg moving at speed 8.6189e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3527e-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. |
7,776 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.6584e-26 kg, speed 9.0139e+05 m/s | A free particle of mass 8.6584e-26 kg moving at speed 9.0139e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 8.4900e-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. |
7,777 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.6270e-26 kg, speed 4.4657e+06 m/s | A free particle of mass 8.6270e-26 kg moving at speed 4.4657e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7199e-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. |
7,778 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.2538e-26 kg, speed 4.0631e+06 m/s | A free particle of mass 8.2538e-26 kg moving at speed 4.0631e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9758e-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. |
7,779 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.0941e-26 kg, speed 9.7677e+06 m/s | A free particle of mass 6.0941e-26 kg moving at speed 9.7677e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.1131e-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. |
7,780 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.7769e-26 kg, speed 6.9069e+06 m/s | A free particle of mass 2.7769e-26 kg moving at speed 6.9069e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.4547e-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. |
7,781 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.6544e-27 kg, speed 1.6097e+06 m/s | A free particle of mass 9.6544e-27 kg moving at speed 1.6097e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.2638e-14 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. |
7,782 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.9831e-26 kg, speed 1.1098e+06 m/s | A free particle of mass 6.9831e-26 kg moving at speed 1.1098e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 8.5497e-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. |
7,783 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.4956e-26 kg, speed 5.7107e+05 m/s | A free particle of mass 5.4956e-26 kg moving at speed 5.7107e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.1113e-14 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. |
7,784 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.7518e-26 kg, speed 4.2231e+06 m/s | A free particle of mass 8.7518e-26 kg moving at speed 4.2231e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7928e-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. |
7,785 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.0947e-27 kg, speed 6.8822e+06 m/s | A free particle of mass 2.0947e-27 kg moving at speed 6.8822e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.5964e-14 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. |
7,786 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.6810e-26 kg, speed 2.2307e+04 m/s | A free particle of mass 1.6810e-26 kg moving at speed 2.2307e+04 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7670e-12 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. |
7,787 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.3926e-28 kg, speed 9.3625e+06 m/s | A free particle of mass 5.3926e-28 kg moving at speed 9.3625e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3124e-13 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. |
7,788 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.4800e-26 kg, speed 5.3929e+06 m/s | A free particle of mass 7.4800e-26 kg moving at speed 5.3929e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.6426e-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. |
7,789 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.9340e-26 kg, speed 3.9609e+06 m/s | A free particle of mass 1.9340e-26 kg moving at speed 3.9609e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 8.6498e-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. |
7,790 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.7297e-26 kg, speed 3.9485e+04 m/s | A free particle of mass 8.7297e-26 kg moving at speed 3.9485e+04 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9223e-13 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. |
7,791 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.3181e-26 kg, speed 3.2670e+06 m/s | A free particle of mass 6.3181e-26 kg moving at speed 3.2670e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.2101e-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. |
7,792 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.7947e-26 kg, speed 2.4435e+06 m/s | A free particle of mass 1.7947e-26 kg moving at speed 2.4435e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5110e-14 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. |
7,793 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.2946e-26 kg, speed 2.3502e+06 m/s | A free particle of mass 5.2946e-26 kg moving at speed 2.3502e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 5.3249e-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. |
7,794 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.0979e-26 kg, speed 9.3532e+06 m/s | A free particle of mass 7.0979e-26 kg moving at speed 9.3532e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 9.9808e-16 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. |
7,795 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.5316e-26 kg, speed 1.6192e+06 m/s | A free particle of mass 5.5316e-26 kg moving at speed 1.6192e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.3978e-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. |
7,796 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.1794e-26 kg, speed 8.1034e+06 m/s | A free particle of mass 4.1794e-26 kg moving at speed 8.1034e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9565e-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. |
7,797 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.7092e-26 kg, speed 7.7838e+06 m/s | A free particle of mass 1.7092e-26 kg moving at speed 7.7838e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.9804e-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. |
7,798 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.6731e-26 kg, speed 6.2542e+05 m/s | A free particle of mass 7.6731e-26 kg moving at speed 6.2542e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3808e-14 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. |
7,799 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.5111e-26 kg, speed 5.3403e+06 m/s | A free particle of mass 4.5111e-26 kg moving at speed 5.3403e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.7505e-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. |
7,800 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.3220e-26 kg, speed 8.2265e+06 m/s | A free particle of mass 4.3220e-26 kg moving at speed 8.2265e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.8636e-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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