How can we prove the kinetic energy equation

[murshiddreamengineer]

TL;DR Summary

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?

 

[Orodruin]

It works individually on each component’s contribution. That is $\int v_1 dv_1 = v_1^2/2$ etc.

 

[murshiddreamengineer]

Thank you. I forgot to think that.