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2,501 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1408 J, W = -621.2 J | A thermodynamic system exchanges heat Q = 1408 J with its surroundings and performs work W = -621.2 J. By the first law, the change in internal energy is ΔU = Q − W = 2029 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. |
2,502 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -453 J, W = -25.19 J | A thermodynamic system exchanges heat Q = -453 J with its surroundings and performs work W = -25.19 J. By the first law, the change in internal energy is ΔU = Q − W = -427.8 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. |
2,503 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 578.1 J, W = -485.8 J | A thermodynamic system exchanges heat Q = 578.1 J with its surroundings and performs work W = -485.8 J. By the first law, the change in internal energy is ΔU = Q − W = 1064 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. |
2,504 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -418 J, W = 255.7 J | A thermodynamic system exchanges heat Q = -418 J with its surroundings and performs work W = 255.7 J. By the first law, the change in internal energy is ΔU = Q − W = -673.7 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. |
2,505 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 982.9 J, W = -49.47 J | A thermodynamic system exchanges heat Q = 982.9 J with its surroundings and performs work W = -49.47 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. |
2,506 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1771 J, W = -730.7 J | A thermodynamic system exchanges heat Q = 1771 J with its surroundings and performs work W = -730.7 J. By the first law, the change in internal energy is ΔU = Q − W = 2502 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. |
2,507 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 254.1 J, W = -783.6 J | A thermodynamic system exchanges heat Q = 254.1 J with its surroundings and performs work W = -783.6 J. By the first law, the change in internal energy is ΔU = Q − W = 1038 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. |
2,508 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -258.7 J, W = -601.9 J | A thermodynamic system exchanges heat Q = -258.7 J with its surroundings and performs work W = -601.9 J. By the first law, the change in internal energy is ΔU = Q − W = 343.2 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. |
2,509 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1004 J, W = 140.2 J | A thermodynamic system exchanges heat Q = 1004 J with its surroundings and performs work W = 140.2 J. By the first law, the change in internal energy is ΔU = Q − W = 864.3 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. |
2,510 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 0.06107 J, W = 273 J | A thermodynamic system exchanges heat Q = 0.06107 J with its surroundings and performs work W = 273 J. By the first law, the change in internal energy is ΔU = Q − W = -272.9 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. |
2,511 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -426.9 J, W = -567.1 J | A thermodynamic system exchanges heat Q = -426.9 J with its surroundings and performs work W = -567.1 J. By the first law, the change in internal energy is ΔU = Q − W = 140.3 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. |
2,512 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1365 J, W = -356.2 J | A thermodynamic system exchanges heat Q = 1365 J with its surroundings and performs work W = -356.2 J. By the first law, the change in internal energy is ΔU = Q − W = 1722 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. |
2,513 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 949.4 J, W = -236.5 J | A thermodynamic system exchanges heat Q = 949.4 J with its surroundings and performs work W = -236.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1186 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. |
2,514 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1638 J, W = 449.3 J | A thermodynamic system exchanges heat Q = 1638 J with its surroundings and performs work W = 449.3 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. |
2,515 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1032 J, W = 7.855 J | A thermodynamic system exchanges heat Q = 1032 J with its surroundings and performs work W = 7.855 J. By the first law, the change in internal energy is ΔU = Q − W = 1025 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. |
2,516 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -329.2 J, W = 511.3 J | A thermodynamic system exchanges heat Q = -329.2 J with its surroundings and performs work W = 511.3 J. By the first law, the change in internal energy is ΔU = Q − W = -840.5 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. |
2,517 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1408 J, W = 46.14 J | A thermodynamic system exchanges heat Q = 1408 J with its surroundings and performs work W = 46.14 J. By the first law, the change in internal energy is ΔU = Q − W = 1362 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. |
2,518 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1290 J, W = 742.8 J | A thermodynamic system exchanges heat Q = 1290 J with its surroundings and performs work W = 742.8 J. By the first law, the change in internal energy is ΔU = Q − W = 546.7 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. |
2,519 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -70.57 J, W = 406.4 J | A thermodynamic system exchanges heat Q = -70.57 J with its surroundings and performs work W = 406.4 J. By the first law, the change in internal energy is ΔU = Q − W = -477 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. |
2,520 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -489.2 J, W = 360.6 J | A thermodynamic system exchanges heat Q = -489.2 J with its surroundings and performs work W = 360.6 J. By the first law, the change in internal energy is ΔU = Q − W = -849.7 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. |
2,521 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1811 J, W = 672.8 J | A thermodynamic system exchanges heat Q = 1811 J with its surroundings and performs work W = 672.8 J. By the first law, the change in internal energy is ΔU = Q − W = 1138 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. |
2,522 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1628 J, W = -40.46 J | A thermodynamic system exchanges heat Q = 1628 J with its surroundings and performs work W = -40.46 J. By the first law, the change in internal energy is ΔU = Q − W = 1669 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. |
2,523 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6934 c | A clock moving at velocity v = 0.6934 c relative to an inertial observer measures a proper time interval Δτ = 1.851 s. The observer measures a dilated interval Δt = γ Δτ = 2.569 s, where γ = 1 / sqrt(1 − v²/c²) = 1.388. 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. |
2,524 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6725 c | A clock moving at velocity v = 0.6725 c relative to an inertial observer measures a proper time interval Δτ = 1.073 s. The observer measures a dilated interval Δt = γ Δτ = 1.45 s, where γ = 1 / sqrt(1 − v²/c²) = 1.351. 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. |
2,525 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8095 c | A clock moving at velocity v = 0.8095 c relative to an inertial observer measures a proper time interval Δτ = 9.813 s. The observer measures a dilated interval Δt = γ Δτ = 16.72 s, where γ = 1 / sqrt(1 − v²/c²) = 1.703. 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. |
2,526 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4605 c | A clock moving at velocity v = 0.4605 c relative to an inertial observer measures a proper time interval Δτ = 5.143 s. The observer measures a dilated interval Δt = γ Δτ = 5.793 s, where γ = 1 / sqrt(1 − v²/c²) = 1.127. 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. |
2,527 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4464 c | A clock moving at velocity v = 0.4464 c relative to an inertial observer measures a proper time interval Δτ = 7.614 s. The observer measures a dilated interval Δt = γ Δτ = 8.509 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. |
2,528 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3671 c | A clock moving at velocity v = 0.3671 c relative to an inertial observer measures a proper time interval Δτ = 4.659 s. The observer measures a dilated interval Δt = γ Δτ = 5.009 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. |
2,529 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5418 c | A clock moving at velocity v = 0.5418 c relative to an inertial observer measures a proper time interval Δτ = 4.958 s. The observer measures a dilated interval Δt = γ Δτ = 5.899 s, where γ = 1 / sqrt(1 − v²/c²) = 1.19. 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. |
2,530 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4071 c | A clock moving at velocity v = 0.4071 c relative to an inertial observer measures a proper time interval Δτ = 2.883 s. The observer measures a dilated interval Δt = γ Δτ = 3.156 s, where γ = 1 / sqrt(1 − v²/c²) = 1.095. 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. |
2,531 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7138 c | A clock moving at velocity v = 0.7138 c relative to an inertial observer measures a proper time interval Δτ = 0.8225 s. The observer measures a dilated interval Δt = γ Δτ = 1.174 s, where γ = 1 / sqrt(1 − v²/c²) = 1.428. 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. |
2,532 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5197 c | A clock moving at velocity v = 0.5197 c relative to an inertial observer measures a proper time interval Δτ = 3.848 s. The observer measures a dilated interval Δt = γ Δτ = 4.504 s, where γ = 1 / sqrt(1 − v²/c²) = 1.17. 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. |
2,533 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4884 c | A clock moving at velocity v = 0.4884 c relative to an inertial observer measures a proper time interval Δτ = 7.362 s. The observer measures a dilated interval Δt = γ Δτ = 8.436 s, where γ = 1 / sqrt(1 − v²/c²) = 1.146. 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. |
2,534 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.684 c | A clock moving at velocity v = 0.684 c relative to an inertial observer measures a proper time interval Δτ = 8.411 s. The observer measures a dilated interval Δt = γ Δτ = 11.53 s, where γ = 1 / sqrt(1 − v²/c²) = 1.371. 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. |
2,535 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.292 c | A clock moving at velocity v = 0.292 c relative to an inertial observer measures a proper time interval Δτ = 4.073 s. The observer measures a dilated interval Δt = γ Δτ = 4.258 s, where γ = 1 / sqrt(1 − v²/c²) = 1.046. 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. |
2,536 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9372 c | A clock moving at velocity v = 0.9372 c relative to an inertial observer measures a proper time interval Δτ = 4.28 s. The observer measures a dilated interval Δt = γ Δτ = 12.27 s, where γ = 1 / sqrt(1 − v²/c²) = 2.867. 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. |
2,537 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8283 c | A clock moving at velocity v = 0.8283 c relative to an inertial observer measures a proper time interval Δτ = 8.948 s. The observer measures a dilated interval Δt = γ Δτ = 15.97 s, where γ = 1 / sqrt(1 − v²/c²) = 1.785. 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. |
2,538 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7821 c | A clock moving at velocity v = 0.7821 c relative to an inertial observer measures a proper time interval Δτ = 9.391 s. The observer measures a dilated interval Δt = γ Δτ = 15.07 s, where γ = 1 / sqrt(1 − v²/c²) = 1.605. 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. |
2,539 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4123 c | A clock moving at velocity v = 0.4123 c relative to an inertial observer measures a proper time interval Δτ = 0.1279 s. The observer measures a dilated interval Δt = γ Δτ = 0.1403 s, where γ = 1 / sqrt(1 − v²/c²) = 1.098. 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. |
2,540 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.673 c | A clock moving at velocity v = 0.673 c relative to an inertial observer measures a proper time interval Δτ = 6.721 s. The observer measures a dilated interval Δt = γ Δτ = 9.087 s, where γ = 1 / sqrt(1 − v²/c²) = 1.352. 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. |
2,541 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4369 c | A clock moving at velocity v = 0.4369 c relative to an inertial observer measures a proper time interval Δτ = 5.227 s. The observer measures a dilated interval Δt = γ Δτ = 5.811 s, where γ = 1 / sqrt(1 − v²/c²) = 1.112. 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. |
2,542 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.675 c | A clock moving at velocity v = 0.675 c relative to an inertial observer measures a proper time interval Δτ = 1.192 s. The observer measures a dilated interval Δt = γ Δτ = 1.616 s, where γ = 1 / sqrt(1 − v²/c²) = 1.355. 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. |
2,543 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4356 c | A clock moving at velocity v = 0.4356 c relative to an inertial observer measures a proper time interval Δτ = 6.95 s. The observer measures a dilated interval Δt = γ Δτ = 7.721 s, where γ = 1 / sqrt(1 − v²/c²) = 1.111. 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. |
2,544 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9309 c | A clock moving at velocity v = 0.9309 c relative to an inertial observer measures a proper time interval Δτ = 7.414 s. The observer measures a dilated interval Δt = γ Δτ = 20.3 s, where γ = 1 / sqrt(1 − v²/c²) = 2.738. 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. |
2,545 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8586 c | A clock moving at velocity v = 0.8586 c relative to an inertial observer measures a proper time interval Δτ = 1.687 s. The observer measures a dilated interval Δt = γ Δτ = 3.291 s, where γ = 1 / sqrt(1 − v²/c²) = 1.95. 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. |
2,546 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4092 c | A clock moving at velocity v = 0.4092 c relative to an inertial observer measures a proper time interval Δτ = 1.991 s. The observer measures a dilated interval Δt = γ Δτ = 2.182 s, where γ = 1 / sqrt(1 − v²/c²) = 1.096. 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. |
2,547 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2702 c | A clock moving at velocity v = 0.2702 c relative to an inertial observer measures a proper time interval Δτ = 8.44 s. The observer measures a dilated interval Δt = γ Δτ = 8.766 s, where γ = 1 / sqrt(1 − v²/c²) = 1.039. 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. |
2,548 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4436 c | A clock moving at velocity v = 0.4436 c relative to an inertial observer measures a proper time interval Δτ = 2.329 s. The observer measures a dilated interval Δt = γ Δτ = 2.598 s, where γ = 1 / sqrt(1 − v²/c²) = 1.116. 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. |
2,549 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.461 c | A clock moving at velocity v = 0.461 c relative to an inertial observer measures a proper time interval Δτ = 4.991 s. The observer measures a dilated interval Δt = γ Δτ = 5.624 s, where γ = 1 / sqrt(1 − v²/c²) = 1.127. 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. |
2,550 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7501 c | A clock moving at velocity v = 0.7501 c relative to an inertial observer measures a proper time interval Δτ = 7.66 s. The observer measures a dilated interval Δt = γ Δτ = 11.58 s, where γ = 1 / sqrt(1 − v²/c²) = 1.512. 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. |
2,551 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6046 c | A clock moving at velocity v = 0.6046 c relative to an inertial observer measures a proper time interval Δτ = 7.563 s. The observer measures a dilated interval Δt = γ Δτ = 9.495 s, where γ = 1 / sqrt(1 − v²/c²) = 1.255. 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. |
2,552 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7808 c | A clock moving at velocity v = 0.7808 c relative to an inertial observer measures a proper time interval Δτ = 9.432 s. The observer measures a dilated interval Δt = γ Δτ = 15.1 s, where γ = 1 / sqrt(1 − v²/c²) = 1.601. 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. |
2,553 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4027 c | A clock moving at velocity v = 0.4027 c relative to an inertial observer measures a proper time interval Δτ = 6.667 s. The observer measures a dilated interval Δt = γ Δτ = 7.283 s, where γ = 1 / sqrt(1 − v²/c²) = 1.092. 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. |
2,554 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9076 c | A clock moving at velocity v = 0.9076 c relative to an inertial observer measures a proper time interval Δτ = 9.875 s. The observer measures a dilated interval Δt = γ Δτ = 23.52 s, where γ = 1 / sqrt(1 − v²/c²) = 2.382. 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. |
2,555 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5347 c | A clock moving at velocity v = 0.5347 c relative to an inertial observer measures a proper time interval Δτ = 1.474 s. The observer measures a dilated interval Δt = γ Δτ = 1.744 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. |
2,556 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4864 c | A clock moving at velocity v = 0.4864 c relative to an inertial observer measures a proper time interval Δτ = 1.222 s. The observer measures a dilated interval Δt = γ Δτ = 1.399 s, where γ = 1 / sqrt(1 − v²/c²) = 1.145. 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. |
2,557 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5159 c | A clock moving at velocity v = 0.5159 c relative to an inertial observer measures a proper time interval Δτ = 2.308 s. The observer measures a dilated interval Δt = γ Δτ = 2.695 s, where γ = 1 / sqrt(1 − v²/c²) = 1.167. 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. |
2,558 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.901 c | A clock moving at velocity v = 0.901 c relative to an inertial observer measures a proper time interval Δτ = 0.2972 s. The observer measures a dilated interval Δt = γ Δτ = 0.6853 s, where γ = 1 / sqrt(1 − v²/c²) = 2.306. 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. |
2,559 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7271 c | A clock moving at velocity v = 0.7271 c relative to an inertial observer measures a proper time interval Δτ = 8.303 s. The observer measures a dilated interval Δt = γ Δτ = 12.09 s, where γ = 1 / sqrt(1 − v²/c²) = 1.457. 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. |
2,560 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5234 c | A clock moving at velocity v = 0.5234 c relative to an inertial observer measures a proper time interval Δτ = 0.2675 s. The observer measures a dilated interval Δt = γ Δτ = 0.3139 s, where γ = 1 / sqrt(1 − v²/c²) = 1.174. 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. |
2,561 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2522 c | A clock moving at velocity v = 0.2522 c relative to an inertial observer measures a proper time interval Δτ = 6.046 s. The observer measures a dilated interval Δt = γ Δτ = 6.248 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. |
2,562 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3779 c | A clock moving at velocity v = 0.3779 c relative to an inertial observer measures a proper time interval Δτ = 3.028 s. The observer measures a dilated interval Δt = γ Δτ = 3.27 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. |
2,563 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.2324e-26 kg, speed 7.6020e+06 m/s | A free particle of mass 6.2324e-26 kg moving at speed 7.6020e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3985e-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. |
2,564 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.6108e-26 kg, speed 5.2000e+06 m/s | A free particle of mass 6.6108e-26 kg moving at speed 5.2000e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9275e-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. |
2,565 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.9219e-26 kg, speed 4.6608e+06 m/s | A free particle of mass 4.9219e-26 kg moving at speed 4.6608e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.8885e-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. |
2,566 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.9357e-26 kg, speed 4.9795e+06 m/s | A free particle of mass 2.9357e-26 kg moving at speed 4.9795e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.5327e-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. |
2,567 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.0480e-26 kg, speed 5.4248e+06 m/s | A free particle of mass 5.0480e-26 kg moving at speed 5.4248e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.4196e-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. |
2,568 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.0948e-26 kg, speed 5.4057e+06 m/s | A free particle of mass 8.0948e-26 kg moving at speed 5.4057e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5142e-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. |
2,569 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.7680e-27 kg, speed 7.6399e+06 m/s | A free particle of mass 1.7680e-27 kg moving at speed 7.6399e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.9055e-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. |
2,570 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.9071e-26 kg, speed 8.8014e+06 m/s | A free particle of mass 9.9071e-26 kg moving at speed 8.8014e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.5990e-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. |
2,571 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.1914e-26 kg, speed 4.5017e+06 m/s | A free particle of mass 1.1914e-26 kg moving at speed 4.5017e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.2354e-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. |
2,572 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.5288e-26 kg, speed 4.4261e+06 m/s | A free particle of mass 8.5288e-26 kg moving at speed 4.4261e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7553e-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. |
2,573 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.5991e-27 kg, speed 1.7695e+06 m/s | A free particle of mass 3.5991e-27 kg moving at speed 1.7695e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0404e-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. |
2,574 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.5941e-26 kg, speed 7.7940e+06 m/s | A free particle of mass 6.5941e-26 kg moving at speed 7.7940e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.2892e-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. |
2,575 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.8470e-26 kg, speed 1.6423e+06 m/s | A free particle of mass 8.8470e-26 kg moving at speed 1.6423e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.5605e-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. |
2,576 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.3069e-26 kg, speed 3.1253e+06 m/s | A free particle of mass 4.3069e-26 kg moving at speed 3.1253e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.9227e-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. |
2,577 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.8909e-26 kg, speed 7.6403e+06 m/s | A free particle of mass 3.8909e-26 kg moving at speed 7.6403e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.2289e-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. |
2,578 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.6886e-26 kg, speed 4.9060e+06 m/s | A free particle of mass 5.6886e-26 kg moving at speed 4.9060e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.3742e-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. |
2,579 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.5379e-26 kg, speed 5.3944e+06 m/s | A free particle of mass 5.5379e-26 kg moving at speed 5.3944e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.2180e-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. |
2,580 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.6982e-26 kg, speed 5.6230e+06 m/s | A free particle of mass 4.6982e-26 kg moving at speed 5.6230e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.5082e-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. |
2,581 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.5632e-26 kg, speed 5.5832e+05 m/s | A free particle of mass 1.5632e-26 kg moving at speed 5.5832e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.5921e-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. |
2,582 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.6340e-26 kg, speed 5.9109e+06 m/s | A free particle of mass 2.6340e-26 kg moving at speed 5.9109e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.2559e-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. |
2,583 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.7780e-26 kg, speed 6.7652e+06 m/s | A free particle of mass 3.7780e-26 kg moving at speed 6.7652e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.5924e-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. |
2,584 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.6647e-26 kg, speed 2.8451e+06 m/s | A free particle of mass 7.6647e-26 kg moving at speed 2.8451e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.0385e-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. |
2,585 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.1433e-27 kg, speed 5.9050e+06 m/s | A free particle of mass 6.1433e-27 kg moving at speed 5.9050e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.8265e-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. |
2,586 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.2821e-26 kg, speed 3.6647e+06 m/s | A free particle of mass 1.2821e-26 kg moving at speed 3.6647e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.4103e-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. |
2,587 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.0956e-26 kg, speed 5.8401e+06 m/s | A free particle of mass 8.0956e-26 kg moving at speed 5.8401e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.4015e-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. |
2,588 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.6520e-26 kg, speed 6.2519e+06 m/s | A free particle of mass 7.6520e-26 kg moving at speed 6.2519e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3851e-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. |
2,589 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.8058e-26 kg, speed 4.2821e+06 m/s | A free particle of mass 3.8058e-26 kg moving at speed 4.2821e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.0659e-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. |
2,590 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.9195e-26 kg, speed 4.1199e+06 m/s | A free particle of mass 9.9195e-26 kg moving at speed 4.1199e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.6214e-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. |
2,591 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.6459e-27 kg, speed 2.4430e+06 m/s | A free particle of mass 5.6459e-27 kg moving at speed 2.4430e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.8039e-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. |
2,592 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.3463e-26 kg, speed 3.1217e+06 m/s | A free particle of mass 2.3463e-26 kg moving at speed 3.1217e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 9.0466e-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. |
2,593 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.4951e-27 kg, speed 6.8802e+06 m/s | A free particle of mass 8.4951e-27 kg moving at speed 6.8802e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.1337e-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. |
2,594 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.6894e-26 kg, speed 1.1456e+06 m/s | A free particle of mass 8.6894e-26 kg moving at speed 1.1456e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 6.6563e-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. |
2,595 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.6979e-26 kg, speed 5.3846e+06 m/s | A free particle of mass 1.6979e-26 kg moving at speed 5.3846e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.2476e-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. |
2,596 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.7387e-27 kg, speed 6.5780e+06 m/s | A free particle of mass 3.7387e-27 kg moving at speed 6.5780e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.6942e-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. |
2,597 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.2248e-26 kg, speed 5.1481e+06 m/s | A free particle of mass 4.2248e-26 kg moving at speed 5.1481e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.0465e-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. |
2,598 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.2118e-26 kg, speed 2.0500e+05 m/s | A free particle of mass 4.2118e-26 kg moving at speed 2.0500e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.6742e-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. |
2,599 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.6971e-26 kg, speed 2.2334e+05 m/s | A free particle of mass 4.6971e-26 kg moving at speed 2.2334e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 6.3163e-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. |
2,600 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.1652e-26 kg, speed 5.7040e+06 m/s | A free particle of mass 3.1652e-26 kg moving at speed 5.7040e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.6701e-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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