Why is Kinetic Energy mv^2/2?

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Why is Kinetic Energy mv^2/2?

Whilst this proof is beyond the scope of A level physics, it is well within the scope of A level Maths as it relies solely on the chain rule.

First let us note that F*x = W where F is force, x is distance and W is work or energy.

However if we have a varying force, we must sum the Force in dx sized chunks, ie. W = int ( F dx) and it is from there we get our proof.

F = ma (from Newton II)

a = dv/dt

F = m*dv/dt

Putting this in our integral:

int(F*dx) = int (m * dv*dx/dt)

We also know that v = dx/dt.

Therefore int(m*dv*dx/dt) = int(m*dv*v) = int(mv dv).

Taking the integral we can see that W = mv²/2 + C where C is dependant on the initial speed we take the work done from. If initial speed is 0 then C also becomes 0 and we get the well known formula.

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