# Difference between momentum and kinetic energy

From a mathematical point of view it seems to be clear what's the difference between momentum and $mv$ and kinetic energy $\frac{1}{2} m v^2$. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities?

From physics I think one can list essentially the following differences:

1. momentum has a direction, kinetic energy not

2. momentum is conserved, kinetic energy not (but energy is)

3. momentum depends linear on velocity, kinetic energy depends quadratically on velocity

I think is is relatively easy to explain the first two points using everyday language, without referring to formulas.

However is there a good way to illustrate the 3. point?

• – Qmechanic Oct 25 '11 at 9:50

As a qualitative understanding, here's an example:

If you shoot a bullet, the rifle recoils with the same momentum as the bullet, but the bullet has a lot more Kinetic energy. Aren't you glad your shoulder is being hit by the rifle stock, and not by the bullet?

• Does the rifle really not have a comparable amount of kinetic energy? It could be that it's just not as harmful as the bullet because it's slower (due to higher mass) and has a much larger surface area that's flat in contrast to the sharp bullet. It really is not clear to me how fast the rifle would go if it weren't held by someone while being fired. – Everyday Astronaut Jan 9 '18 at 12:29
• @DrDoolittle: momentum $= mv$, right? and kinetic energy $=mv^2/2$, right? OK, so increase the mass 100 times and divide the velocity by 100. Same momentum, right? $100mv/100 = mv$ But what does it do to the kinetic energy? $100m(v/100)^2/2 = mv^2/100/2$. So by increasing the mass 100 times, for the same momentum, we've cut the kinetic energy by a factor of 100. – Mike Dunlavey Jan 9 '18 at 17:25
• Correct. So it's all about point 2 of the OP. My confusion persists, however, since I wonder why kinetic energy is "dangerous" but momentum isn't. That's probably a philosophical question... I guess I must stop subconsciously thinking that momentum and energy are somehow similar ... they are not. – Everyday Astronaut Jan 9 '18 at 20:06
• @DrDoolittle: Momentum is conserved, and energy is conserved, of the ensemble, bullet and gun together. The momentum before firing is zero, and so it is after, split into equal parts going in opposite directions. The energy before firing is non-zero, it is chemical energy. Firing changes it into kinetic (and heat) energy, shared by both bullet and gun, but more in the bullet because it has higher speed. Note that momentum has a direction, but energy does not. It's positive either way, because it's squared. – Mike Dunlavey Jan 9 '18 at 21:59
• The essence of the difference seems to lie in the emphasis on velocity in the energy equation. I think the OP is attempting to conceptualize how this can be. I know I am. – Tom Russell Jan 31 '18 at 8:15

As a matter of fact, the second point couldn't be more wrong. Momentum is far more ubiquitous than kinetic energy since it is a conserved quantity of every physical system that is translationally invariant.

With respect to your question, user Gerard gave an explanation as intuitive as it gets here

• Yes, sorry. I too narrowly thought about elementary macroscopic physics. What I had in mind was that kinetic energy can be converted to thermic energy, however that there is no analogue of momentum in thermodynamics, but perhaps I am wrong and there is also momentum in thermodynamics... – martin Oct 25 '11 at 10:37

## protected by Qmechanic♦May 24 '14 at 17:42

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