Why is the equation for Kinetic Energy 1/2mv^2 when the equation for Momentum is mv?

In: Physics

They are completely different measurements that describe different physical properties.

The momentum describes how much force wee need to apply and for how long we need to apply it in order to change the objects velocity a certain amount (F•t) . Just like forces, the momentum has a direction vector.

The kinetic energy describes how much work (energy) is needed to stop the object’s movement. The kinetic energy has no direction.

Note that the fact that these two equations are different explains why [Newton’s Cradle](https://en.wikipedia.org/wiki/Newton%27s_cradle) works the way it does. The only way to conserve both momentum and kinetic energy is to also conserve mass, because both *v* and *v*^2 are involved. That’s why, if you drop one ball on one end, one ball bounces out the other, and not two or more balls at a lower speed.

Kinetic energy comes from applying a certain force over a certain distance. Momentum comes from applying a certain force over a certain time.

Now, lets say you’re at rest and a force accelerates you for 10 seconds. You will have covered some distance. Now if the same force accelerates you again for 10 seconds, you will have covered a greater distance the second time, because you were going faster the entire time. So your change in momentum from 0-10 and 10-20 is the same, since it’s the same force over the same time. But your change in kinetic energy is bigger for the second push since over those 10 seconds you were going faster and covered more distance. This gets translated into a v^2.

First off, both of those are approximations which work well on human scales but less well for relativistic velocities and massless particles.

That said: we don’t know if there *is* a “why”, let alone what it might be. We’ve simply observed that that’s the way the universe works. Unless a Creator of the universe makes itself available for questioning it’s entirely possible we’ll never know if there is a “why”.