How does mass relate to kinetic energy?

Question

I was working on homework and I had to list what kinetic energy depends on. Interestingly, it turns out it's also based on mass, not only speed. Does anyone have an explanation?

Answer 1

Hopefully this will help you build some intuition: Consider two objects with masses $M$ and $m$, where $M$ is much greater than $m$. One object might be a large boulder with mass $M$ and the other a golf ball with mass $m$. When bringing both objects from rest to the same velocity $v$, which object do you expect to require more energy? The energy you have put into the object so that it reaches velocity $v$, the work, is the kinetic energy of the object.

The derivation on the kinetic energy wiki page might also help.

Answer 2

Maple's answer already gives a good intuition, so I'll just give another example. Gravitational acceleration is independent of the mass of the object, so objects with different masses will move in the same way when they are free falling. Therefore, if we drop two objects from the same height, they hit the ground with the same speed, but with different kinetic energies (and also different momenta).

Given this, suppose you are standing on the ground and something falls on you from the first floor of the building next to you. From the same height, which causes more damage: a ping pong ball or a piano? Both hit you with the same speed, but the piano has a lot more energy, so it causes a lot more damage, there is a lot more sound dispersed, and so on.

Out of the intuition realm, it might be worth recalling that the formula for kinetic energy $K$ is $$K = \frac{m v^2}{2},$$ where $m$ is the object's mass and $v$ is its velocity.

Answer 3

If a moving baseball, bullet, atom, whatever has a certain amount of kinetic energy because of its motion then we should expect that two baseballs, bullets, atoms, etc. moving with the same speed should have twice that amount of kinetic energy.

Mass here is just a measure of how much stuff is moving. It would be surprising if the kinetic energy of an object were not proportional to its mass.

Answer 4

Interestingly, it turns out it's also based on mass, not only speed. Does anyone have an explanation?

Yes, the initial explanation is that kinetic energy is simply defined that way: $$ K \equiv \frac{1}{2}mv^2\;. $$

If you mean to ask, why was it defined that way, it was defined that was because it is a very useful definition.

It is a useful definition because, in the elementary case, we see that: $$ \frac{d}{dt}\frac{\partial K}{\partial v} = ma = F\;,\tag{1} $$ where the last equality comes from Newton's second law.

But, more generally, a relation like Eq. (1) continues to hold even for complex systems of multiple particles, and even when the kinetic energy is written in terms of generalized coordinates $q_i$ and velocites $\dot q_i$. The more general expression is: $$ \frac{d}{dt}\frac{\partial K}{\partial \dot q_i} - \frac{\partial K}{\partial q_i} = F_i\;, \tag{2} $$ where $F_i$ is the generalized force that determines the work done when one of the $q_i$ changes.