I How can we prove the kinetic energy equation

AI Thread Summary
The discussion focuses on proving the kinetic energy equation through the work-energy principle, stating that the work done equals the change in kinetic energy in a mechanical system. It outlines the mathematical derivation of work done, starting from the force applied to displacement and integrating over velocity. The conversation highlights that when velocity and its differential are aligned, the standard kinetic energy equation of 1/2mv^2 holds true. It also addresses scenarios where velocity components are in different directions, suggesting that the general kinetic energy equation can still be validated by analyzing each component separately. The conclusion emphasizes the importance of considering directional components in kinetic energy calculations.
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does the kinetic energy equation always work? it has proved for only when acceleration and and velocity are in the same direction.
Proof of kinetic energy
work done equals a change in kinetic energy in a mechanical system.
δW = F.ds
W = ∫F.ds
W = m∫a.ds
W = m∫(dv/dt).ds
W = m∫v.dv

here if v and dv are in the same direction the change in kinetic energy will be the usual equation. what happens if both are in different directions how can we prove that the general equation for kinetic energy will be 1/2mv^2?
 
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It works individually on each component’s contribution. That is ##\int v_1 dv_1 = v_1^2/2## etc.
 
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Thank you. I forgot to think that.
 
Hello everyone, Consider the problem in which a car is told to travel at 30 km/h for L kilometers and then at 60 km/h for another L kilometers. Next, you are asked to determine the average speed. My question is: although we know that the average speed in this case is the harmonic mean of the two speeds, is it also possible to state that the average speed over this 2L-kilometer stretch can be obtained as a weighted average of the two speeds? Best regards, DaTario
This has been discussed many times on PF, and will likely come up again, so the video might come handy. Previous threads: https://www.physicsforums.com/threads/is-a-treadmill-incline-just-a-marketing-gimmick.937725/ https://www.physicsforums.com/threads/work-done-running-on-an-inclined-treadmill.927825/ https://www.physicsforums.com/threads/how-do-we-calculate-the-energy-we-used-to-do-something.1052162/

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