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20,132,601 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1576 J, W = -39.05 J | A thermodynamic system exchanges heat Q = 1576 J with its surroundings and performs work W = -39.05 J. By the first law, the change in internal energy is ΔU = Q − W = 1615 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. |
20,132,602 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = -208 J, W = -336.6 J | A thermodynamic system exchanges heat Q = -208 J with its surroundings and performs work W = -336.6 J. By the first law, the change in internal energy is ΔU = Q − W = 128.6 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. |
20,132,603 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1911 J, W = -422.2 J | A thermodynamic system exchanges heat Q = 1911 J with its surroundings and performs work W = -422.2 J. By the first law, the change in internal energy is ΔU = Q − W = 2334 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. |
20,132,604 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 116.4 J, W = 154.3 J | A thermodynamic system exchanges heat Q = 116.4 J with its surroundings and performs work W = 154.3 J. By the first law, the change in internal energy is ΔU = Q − W = -37.89 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. |
20,132,605 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1852 J, W = -728.7 J | A thermodynamic system exchanges heat Q = 1852 J with its surroundings and performs work W = -728.7 J. By the first law, the change in internal energy is ΔU = Q − W = 2580 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. |
20,132,606 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1389 J, W = -271.5 J | A thermodynamic system exchanges heat Q = 1389 J with its surroundings and performs work W = -271.5 J. By the first law, the change in internal energy is ΔU = Q − W = 1660 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. |
20,132,607 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 273.9 J, W = 780.9 J | A thermodynamic system exchanges heat Q = 273.9 J with its surroundings and performs work W = 780.9 J. By the first law, the change in internal energy is ΔU = Q − W = -507 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. |
20,132,608 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 159.4 J, W = -361 J | A thermodynamic system exchanges heat Q = 159.4 J with its surroundings and performs work W = -361 J. By the first law, the change in internal energy is ΔU = Q − W = 520.4 J. The first law is a statement of conservation of energy applied to thermodynamic systems; internal energy is a state function. | ΔU = Q - W | mechanical_energy | Apply the first law of thermodynamics to compute the change in internal energy. |
20,132,609 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1929 J, W = 267.9 J | A thermodynamic system exchanges heat Q = 1929 J with its surroundings and performs work W = 267.9 J. By the first law, the change in internal energy is ΔU = Q − W = 1661 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. |
20,132,610 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1751 J, W = -22.67 J | A thermodynamic system exchanges heat Q = 1751 J with its surroundings and performs work W = -22.67 J. By the first law, the change in internal energy is ΔU = Q − W = 1774 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. |
20,132,611 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1846 J, W = -594.9 J | A thermodynamic system exchanges heat Q = 1846 J with its surroundings and performs work W = -594.9 J. By the first law, the change in internal energy is ΔU = Q − W = 2441 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. |
20,132,612 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 81.06 J, W = -313 J | A thermodynamic system exchanges heat Q = 81.06 J with its surroundings and performs work W = -313 J. By the first law, the change in internal energy is ΔU = Q − W = 394.1 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. |
20,132,613 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1709 J, W = 592.2 J | A thermodynamic system exchanges heat Q = 1709 J with its surroundings and performs work W = 592.2 J. By the first law, the change in internal energy is ΔU = Q − W = 1117 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. |
20,132,614 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 1951 J, W = -174.6 J | A thermodynamic system exchanges heat Q = 1951 J with its surroundings and performs work W = -174.6 J. By the first law, the change in internal energy is ΔU = Q − W = 2126 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. |
20,132,615 | physics | thermodynamics | first_law | 6 | worked_example | First law of thermodynamics: Q = 708.5 J, W = 733.2 J | A thermodynamic system exchanges heat Q = 708.5 J with its surroundings and performs work W = 733.2 J. By the first law, the change in internal energy is ΔU = Q − W = -24.71 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. |
20,132,616 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7053 c | A clock moving at velocity v = 0.7053 c relative to an inertial observer measures a proper time interval Δτ = 2.28 s. The observer measures a dilated interval Δt = γ Δτ = 3.217 s, where γ = 1 / sqrt(1 − v²/c²) = 1.411. 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. |
20,132,617 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9201 c | A clock moving at velocity v = 0.9201 c relative to an inertial observer measures a proper time interval Δτ = 6.292 s. The observer measures a dilated interval Δt = γ Δτ = 16.06 s, where γ = 1 / sqrt(1 − v²/c²) = 2.552. 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. |
20,132,618 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1363 c | A clock moving at velocity v = 0.1363 c relative to an inertial observer measures a proper time interval Δτ = 9.187 s. The observer measures a dilated interval Δt = γ Δτ = 9.274 s, where γ = 1 / sqrt(1 − v²/c²) = 1.009. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,619 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2354 c | A clock moving at velocity v = 0.2354 c relative to an inertial observer measures a proper time interval Δτ = 7.962 s. The observer measures a dilated interval Δt = γ Δτ = 8.192 s, where γ = 1 / sqrt(1 − v²/c²) = 1.029. 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. |
20,132,620 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9356 c | A clock moving at velocity v = 0.9356 c relative to an inertial observer measures a proper time interval Δτ = 3.921 s. The observer measures a dilated interval Δt = γ Δτ = 11.11 s, where γ = 1 / sqrt(1 − v²/c²) = 2.833. 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. |
20,132,621 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3348 c | A clock moving at velocity v = 0.3348 c relative to an inertial observer measures a proper time interval Δτ = 7.773 s. The observer measures a dilated interval Δt = γ Δτ = 8.25 s, where γ = 1 / sqrt(1 − v²/c²) = 1.061. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,622 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6779 c | A clock moving at velocity v = 0.6779 c relative to an inertial observer measures a proper time interval Δτ = 0.04326 s. The observer measures a dilated interval Δt = γ Δτ = 0.05885 s, where γ = 1 / sqrt(1 − v²/c²) = 1.36. 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. |
20,132,623 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9432 c | A clock moving at velocity v = 0.9432 c relative to an inertial observer measures a proper time interval Δτ = 3.827 s. The observer measures a dilated interval Δt = γ Δτ = 11.52 s, where γ = 1 / sqrt(1 − v²/c²) = 3.01. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,624 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4272 c | A clock moving at velocity v = 0.4272 c relative to an inertial observer measures a proper time interval Δτ = 0.5881 s. The observer measures a dilated interval Δt = γ Δτ = 0.6505 s, where γ = 1 / sqrt(1 − v²/c²) = 1.106. 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. |
20,132,625 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4019 c | A clock moving at velocity v = 0.4019 c relative to an inertial observer measures a proper time interval Δτ = 7.697 s. The observer measures a dilated interval Δt = γ Δτ = 8.406 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. |
20,132,626 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.1439 c | A clock moving at velocity v = 0.1439 c relative to an inertial observer measures a proper time interval Δτ = 9.268 s. The observer measures a dilated interval Δt = γ Δτ = 9.366 s, where γ = 1 / sqrt(1 − v²/c²) = 1.011. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,627 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6601 c | A clock moving at velocity v = 0.6601 c relative to an inertial observer measures a proper time interval Δτ = 4.167 s. The observer measures a dilated interval Δt = γ Δτ = 5.547 s, where γ = 1 / sqrt(1 − v²/c²) = 1.331. 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. |
20,132,628 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2181 c | A clock moving at velocity v = 0.2181 c relative to an inertial observer measures a proper time interval Δτ = 1.379 s. The observer measures a dilated interval Δt = γ Δτ = 1.413 s, where γ = 1 / sqrt(1 − v²/c²) = 1.025. 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. |
20,132,629 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6126 c | A clock moving at velocity v = 0.6126 c relative to an inertial observer measures a proper time interval Δτ = 6.638 s. The observer measures a dilated interval Δt = γ Δτ = 8.398 s, where γ = 1 / sqrt(1 − v²/c²) = 1.265. 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. |
20,132,630 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.318 c | A clock moving at velocity v = 0.318 c relative to an inertial observer measures a proper time interval Δτ = 2.505 s. The observer measures a dilated interval Δt = γ Δτ = 2.643 s, where γ = 1 / sqrt(1 − v²/c²) = 1.055. 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. |
20,132,631 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4402 c | A clock moving at velocity v = 0.4402 c relative to an inertial observer measures a proper time interval Δτ = 1.824 s. The observer measures a dilated interval Δt = γ Δτ = 2.031 s, where γ = 1 / sqrt(1 − v²/c²) = 1.114. 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. |
20,132,632 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5005 c | A clock moving at velocity v = 0.5005 c relative to an inertial observer measures a proper time interval Δτ = 9.653 s. The observer measures a dilated interval Δt = γ Δτ = 11.15 s, where γ = 1 / sqrt(1 − v²/c²) = 1.155. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,633 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2479 c | A clock moving at velocity v = 0.2479 c relative to an inertial observer measures a proper time interval Δτ = 3.592 s. The observer measures a dilated interval Δt = γ Δτ = 3.708 s, where γ = 1 / sqrt(1 − v²/c²) = 1.032. 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. |
20,132,634 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8444 c | A clock moving at velocity v = 0.8444 c relative to an inertial observer measures a proper time interval Δτ = 5.952 s. The observer measures a dilated interval Δt = γ Δτ = 11.11 s, where γ = 1 / sqrt(1 − v²/c²) = 1.867. 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. |
20,132,635 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5538 c | A clock moving at velocity v = 0.5538 c relative to an inertial observer measures a proper time interval Δτ = 5.459 s. The observer measures a dilated interval Δt = γ Δτ = 6.556 s, where γ = 1 / sqrt(1 − v²/c²) = 1.201. 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. |
20,132,636 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.758 c | A clock moving at velocity v = 0.758 c relative to an inertial observer measures a proper time interval Δτ = 8.089 s. The observer measures a dilated interval Δt = γ Δτ = 12.4 s, where γ = 1 / sqrt(1 − v²/c²) = 1.533. 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. |
20,132,637 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3725 c | A clock moving at velocity v = 0.3725 c relative to an inertial observer measures a proper time interval Δτ = 0.7692 s. The observer measures a dilated interval Δt = γ Δτ = 0.8288 s, where γ = 1 / sqrt(1 − v²/c²) = 1.078. Time dilation is a direct consequence of the invariance of the spacetime interval. | Δt = γ Δτ; γ = 1 / sqrt(1 - v²/c²) | classical kinematics | Calculate the time-dilation factor and the dilated time interval. |
20,132,638 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6366 c | A clock moving at velocity v = 0.6366 c relative to an inertial observer measures a proper time interval Δτ = 8.343 s. The observer measures a dilated interval Δt = γ Δτ = 10.82 s, where γ = 1 / sqrt(1 − v²/c²) = 1.297. 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. |
20,132,639 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8038 c | A clock moving at velocity v = 0.8038 c relative to an inertial observer measures a proper time interval Δτ = 0.2598 s. The observer measures a dilated interval Δt = γ Δτ = 0.4366 s, where γ = 1 / sqrt(1 − v²/c²) = 1.681. 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. |
20,132,640 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5305 c | A clock moving at velocity v = 0.5305 c relative to an inertial observer measures a proper time interval Δτ = 4.697 s. The observer measures a dilated interval Δt = γ Δτ = 5.542 s, where γ = 1 / sqrt(1 − v²/c²) = 1.18. 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. |
20,132,641 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6937 c | A clock moving at velocity v = 0.6937 c relative to an inertial observer measures a proper time interval Δτ = 4.914 s. The observer measures a dilated interval Δt = γ Δτ = 6.823 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. |
20,132,642 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6015 c | A clock moving at velocity v = 0.6015 c relative to an inertial observer measures a proper time interval Δτ = 8.591 s. The observer measures a dilated interval Δt = γ Δτ = 10.75 s, where γ = 1 / sqrt(1 − v²/c²) = 1.252. 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. |
20,132,643 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6552 c | A clock moving at velocity v = 0.6552 c relative to an inertial observer measures a proper time interval Δτ = 4.507 s. The observer measures a dilated interval Δt = γ Δτ = 5.966 s, where γ = 1 / sqrt(1 − v²/c²) = 1.324. 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. |
20,132,644 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4108 c | A clock moving at velocity v = 0.4108 c relative to an inertial observer measures a proper time interval Δτ = 9.319 s. The observer measures a dilated interval Δt = γ Δτ = 10.22 s, where γ = 1 / sqrt(1 − v²/c²) = 1.097. 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. |
20,132,645 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7573 c | A clock moving at velocity v = 0.7573 c relative to an inertial observer measures a proper time interval Δτ = 7.638 s. The observer measures a dilated interval Δt = γ Δτ = 11.69 s, where γ = 1 / sqrt(1 − v²/c²) = 1.531. 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. |
20,132,646 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2461 c | A clock moving at velocity v = 0.2461 c relative to an inertial observer measures a proper time interval Δτ = 6.228 s. The observer measures a dilated interval Δt = γ Δτ = 6.426 s, where γ = 1 / sqrt(1 − v²/c²) = 1.032. 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. |
20,132,647 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.2842 c | A clock moving at velocity v = 0.2842 c relative to an inertial observer measures a proper time interval Δτ = 7.692 s. The observer measures a dilated interval Δt = γ Δτ = 8.023 s, where γ = 1 / sqrt(1 − v²/c²) = 1.043. 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. |
20,132,648 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.5717 c | A clock moving at velocity v = 0.5717 c relative to an inertial observer measures a proper time interval Δτ = 6.873 s. The observer measures a dilated interval Δt = γ Δτ = 8.377 s, where γ = 1 / sqrt(1 − v²/c²) = 1.219. 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. |
20,132,649 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.8306 c | A clock moving at velocity v = 0.8306 c relative to an inertial observer measures a proper time interval Δτ = 3.976 s. The observer measures a dilated interval Δt = γ Δτ = 7.139 s, where γ = 1 / sqrt(1 − v²/c²) = 1.796. 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. |
20,132,650 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9265 c | A clock moving at velocity v = 0.9265 c relative to an inertial observer measures a proper time interval Δτ = 2.984 s. The observer measures a dilated interval Δt = γ Δτ = 7.931 s, where γ = 1 / sqrt(1 − v²/c²) = 2.658. 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. |
20,132,651 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.6043 c | A clock moving at velocity v = 0.6043 c relative to an inertial observer measures a proper time interval Δτ = 5.291 s. The observer measures a dilated interval Δt = γ Δτ = 6.641 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. |
20,132,652 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.7557 c | A clock moving at velocity v = 0.7557 c relative to an inertial observer measures a proper time interval Δτ = 5.843 s. The observer measures a dilated interval Δt = γ Δτ = 8.921 s, where γ = 1 / sqrt(1 − v²/c²) = 1.527. 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. |
20,132,653 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.3246 c | A clock moving at velocity v = 0.3246 c relative to an inertial observer measures a proper time interval Δτ = 0.9439 s. The observer measures a dilated interval Δt = γ Δτ = 0.998 s, where γ = 1 / sqrt(1 − v²/c²) = 1.057. 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. |
20,132,654 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.4634 c | A clock moving at velocity v = 0.4634 c relative to an inertial observer measures a proper time interval Δτ = 9.553 s. The observer measures a dilated interval Δt = γ Δτ = 10.78 s, where γ = 1 / sqrt(1 − v²/c²) = 1.128. 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. |
20,132,655 | physics | relativity | time_dilation | 8 | worked_example | Time dilation at v = 0.9271 c | A clock moving at velocity v = 0.9271 c relative to an inertial observer measures a proper time interval Δτ = 2.597 s. The observer measures a dilated interval Δt = γ Δτ = 6.929 s, where γ = 1 / sqrt(1 − v²/c²) = 2.668. 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. |
20,132,656 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.5063e-26 kg, speed 6.5602e+06 m/s | A free particle of mass 4.5063e-26 kg moving at speed 6.5602e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.2414e-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. |
20,132,657 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.2352e-26 kg, speed 1.4435e+06 m/s | A free particle of mass 5.2352e-26 kg moving at speed 1.4435e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 8.7682e-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. |
20,132,658 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.7758e-26 kg, speed 3.8285e+06 m/s | A free particle of mass 8.7758e-26 kg moving at speed 3.8285e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9722e-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. |
20,132,659 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.1710e-26 kg, speed 2.4887e+05 m/s | A free particle of mass 4.1710e-26 kg moving at speed 2.4887e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 6.3833e-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. |
20,132,660 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.4214e-26 kg, speed 5.0921e+05 m/s | A free particle of mass 7.4214e-26 kg moving at speed 5.0921e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7534e-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. |
20,132,661 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.5897e-26 kg, speed 5.8594e+06 m/s | A free particle of mass 2.5897e-26 kg moving at speed 5.8594e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.3667e-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. |
20,132,662 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.4691e-26 kg, speed 6.9230e+06 m/s | A free particle of mass 3.4691e-26 kg moving at speed 6.9230e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.7589e-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. |
20,132,663 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.2491e-26 kg, speed 6.8619e+06 m/s | A free particle of mass 9.2491e-26 kg moving at speed 6.8619e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0440e-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. |
20,132,664 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.9589e-26 kg, speed 7.1010e+06 m/s | A free particle of mass 7.9589e-26 kg moving at speed 7.1010e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.1724e-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. |
20,132,665 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.7178e-26 kg, speed 9.0880e+06 m/s | A free particle of mass 9.7178e-26 kg moving at speed 9.0880e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.5028e-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. |
20,132,666 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.7564e-26 kg, speed 2.8019e+06 m/s | A free particle of mass 9.7564e-26 kg moving at speed 2.8019e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.4239e-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. |
20,132,667 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.3588e-26 kg, speed 4.7628e+06 m/s | A free particle of mass 1.3588e-26 kg moving at speed 4.7628e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0239e-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. |
20,132,668 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.0164e-26 kg, speed 7.7603e+06 m/s | A free particle of mass 3.0164e-26 kg moving at speed 7.7603e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.8307e-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. |
20,132,669 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.0884e-26 kg, speed 1.0928e+06 m/s | A free particle of mass 6.0884e-26 kg moving at speed 1.0928e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 9.9588e-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. |
20,132,670 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.9756e-26 kg, speed 7.0718e+06 m/s | A free particle of mass 6.9756e-26 kg moving at speed 7.0718e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3432e-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. |
20,132,671 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.1701e-26 kg, speed 5.3111e+06 m/s | A free particle of mass 1.1701e-26 kg moving at speed 5.3111e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0662e-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. |
20,132,672 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.5428e-26 kg, speed 8.6081e+06 m/s | A free particle of mass 4.5428e-26 kg moving at speed 8.6081e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.6944e-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. |
20,132,673 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.1284e-26 kg, speed 4.2147e+06 m/s | A free particle of mass 5.1284e-26 kg moving at speed 4.2147e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.0655e-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. |
20,132,674 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.5608e-26 kg, speed 6.5987e+06 m/s | A free particle of mass 1.5608e-26 kg moving at speed 6.5987e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 6.4336e-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. |
20,132,675 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.1724e-26 kg, speed 4.5536e+06 m/s | A free particle of mass 8.1724e-26 kg moving at speed 4.5536e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7806e-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. |
20,132,676 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.5020e-26 kg, speed 3.2232e+06 m/s | A free particle of mass 1.5020e-26 kg moving at speed 3.2232e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.3687e-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. |
20,132,677 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 1.2391e-26 kg, speed 2.3849e+06 m/s | A free particle of mass 1.2391e-26 kg moving at speed 2.3849e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.2422e-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. |
20,132,678 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.8464e-26 kg, speed 8.0050e+06 m/s | A free particle of mass 2.8464e-26 kg moving at speed 8.0050e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.9080e-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. |
20,132,679 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.2632e-26 kg, speed 7.8041e+06 m/s | A free particle of mass 8.2632e-26 kg moving at speed 7.8041e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0275e-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. |
20,132,680 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.5142e-26 kg, speed 4.8750e+06 m/s | A free particle of mass 5.5142e-26 kg moving at speed 4.8750e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.4649e-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. |
20,132,681 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.5478e-26 kg, speed 9.8115e+06 m/s | A free particle of mass 3.5478e-26 kg moving at speed 9.8115e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.9036e-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. |
20,132,682 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.1505e-27 kg, speed 8.9422e+06 m/s | A free particle of mass 4.1505e-27 kg moving at speed 8.9422e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.7853e-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. |
20,132,683 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.7866e-26 kg, speed 3.1823e+06 m/s | A free particle of mass 8.7866e-26 kg moving at speed 3.1823e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.3697e-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. |
20,132,684 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.1865e-27 kg, speed 9.8651e+06 m/s | A free particle of mass 9.1865e-27 kg moving at speed 9.8651e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.3115e-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. |
20,132,685 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.9605e-26 kg, speed 8.6128e+06 m/s | A free particle of mass 2.9605e-26 kg moving at speed 8.6128e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.5987e-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. |
20,132,686 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.3348e-26 kg, speed 3.6690e+06 m/s | A free particle of mass 2.3348e-26 kg moving at speed 3.6690e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 7.7348e-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. |
20,132,687 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.2777e-26 kg, speed 9.9274e+06 m/s | A free particle of mass 4.2777e-26 kg moving at speed 9.9274e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5603e-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. |
20,132,688 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 6.9870e-26 kg, speed 9.8024e+06 m/s | A free particle of mass 6.9870e-26 kg moving at speed 9.8024e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 9.6745e-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. |
20,132,689 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.7576e-26 kg, speed 6.0909e+06 m/s | A free particle of mass 5.7576e-26 kg moving at speed 6.0909e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.8894e-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. |
20,132,690 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.6432e-26 kg, speed 6.6096e+06 m/s | A free particle of mass 3.6432e-26 kg moving at speed 6.6096e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 2.7516e-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. |
20,132,691 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 4.7632e-26 kg, speed 8.7886e+06 m/s | A free particle of mass 4.7632e-26 kg moving at speed 8.7886e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5828e-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. |
20,132,692 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.7014e-26 kg, speed 2.4148e+06 m/s | A free particle of mass 7.7014e-26 kg moving at speed 2.4148e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 3.5629e-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. |
20,132,693 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 2.4706e-26 kg, speed 1.7407e+06 m/s | A free particle of mass 2.4706e-26 kg moving at speed 1.7407e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5408e-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. |
20,132,694 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.2201e-26 kg, speed 4.5712e+06 m/s | A free particle of mass 9.2201e-26 kg moving at speed 4.5712e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.5721e-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. |
20,132,695 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.2353e-26 kg, speed 2.3710e+06 m/s | A free particle of mass 3.2353e-26 kg moving at speed 2.3710e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 8.6379e-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. |
20,132,696 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 3.4289e-26 kg, speed 4.5053e+05 m/s | A free particle of mass 3.4289e-26 kg moving at speed 4.5053e+05 m/s has de Broglie wavelength λ = h / p = h / (m v) = 4.2893e-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. |
20,132,697 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 8.2780e-26 kg, speed 8.7741e+04 m/s | A free particle of mass 8.2780e-26 kg moving at speed 8.7741e+04 m/s has de Broglie wavelength λ = h / p = h / (m v) = 9.1228e-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. |
20,132,698 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 7.9536e-26 kg, speed 8.3051e+06 m/s | A free particle of mass 7.9536e-26 kg moving at speed 8.3051e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0031e-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. |
20,132,699 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 9.8089e-26 kg, speed 6.2961e+06 m/s | A free particle of mass 9.8089e-26 kg moving at speed 6.2961e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.0729e-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. |
20,132,700 | physics | quantum | de_broglie | 7 | worked_example | de Broglie wavelength of particle mass 5.9926e-26 kg, speed 9.7856e+06 m/s | A free particle of mass 5.9926e-26 kg moving at speed 9.7856e+06 m/s has de Broglie wavelength λ = h / p = h / (m v) = 1.1299e-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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