Kinetic energy and momentum.

Ek = 0.5mv^2
P=mv

m=1, v=1
Ek = 0.5

m=1, v=2
Ek = 2

Accelerating a 1 kg object to 1m/s requires 0.5 joules whilst accelerating it a further 1m/s to 2m/s requires an additional 1.5 joules. Why is this?

>>1
Energy is acceleration times mass times distance. Suppose that over some distance you accelerate this mass from a m/s to b m/s. Then b^2=a^2+2*acceleration*distance. Multiply it by mass and you get mass*b^2=mass*a^2+2*acceleration*mass*distance=mass*a^2+2*energy.
Therefore, energy=(1/2)mass*(b^2-a^2)