# Conservation of momentum and work

Consider the following scenario:

I am alone in space with a ball. I threw the ball at a certain speed such that i am propelled backwards (like a rocket is) as to conserve momentum. Now, since my momentum changed, so does my kinetic energy. By Work-Kinetic Energy Theorem, if there is a change in kinetic energy, there should also be work that caused it. Who did work on me?

What about the force? My body’s center of mass can not accelerate if there is no external force (Newton’s second law). What applied force on my body (if there’s any)?

I am just not a fan of “conservation of such and such says so”. I just want some cause to the change of my velocity since this is still in the realm of newtonian dynamics’ validity.

You are correct that you have to explain the "mechanism" with which the general expectation of conservation laws would be satisfied. Or, in other words, the expectations of the conservation laws should play well the specific laws of dynamics, otherwise, we really wouldn't have a reason to think the conservation laws are true in the first place.

• When you throw the ball, you apply a force on the ball. The forces between you and the ball are of such nature that they follow the third law of Newton. This means that the ball will apply a force of the same magnitude on you in the opposite direction. This would be the dynamic cause of you starting to move backward.

• The work part is interesting. Initially, both you and the ball were at rest. Now, both you and the ball are moving. So, where did all that kinetic energy come from? It'd come from the chemical energy of your muscles which got converted to kinetic energy when you pushed the ball. So, as the answer by @PradyothShandilya says, you did the work.

As an analogy, think about this: imagine two massive boxes sitting close to each other. One box has a compressed spring attached to it on the outside of one of its walls (the wall that is close to the other box). The spring is kept compressed by a string. Now, if I cut this string (which would take negligible work), the spring would start expanding and would push away the other box and also push back the box that it was attached to. It is easy to see in this case that the spring did the work (using its potential energy) to put both the boxes in motion. Your muscles do the same thing.

• Since the ball actually did push me backwards, is it safe to say that the ball did work on me? I can not find it to make sense that I did work on myself If I consider my body to be the system. If I am alone in space without a ball (or anything I can throw), I have no means of moving my center of mass without pushing unto anything, even though I have potential energy in my body.
– Lyle
Commented May 26, 2020 at 2:26
• @Lyle No, the ball does negative work on you because it gains kinetic energy from its interaction with you. This negative work is more than compensated by the work of your muscles so as to overall increase your kinetic energy. It's true that you cannot set yourself into such a motion that your center of mass moves without a ball to throw. But that's related to momentum. The ball applies force on you so as to change your total momentum. But you can absolutely move your arms around without a ball so that your kinetic energy increases while keeping your center of mass at rest. Makes sense?
– user87745
Commented May 26, 2020 at 12:50
• @DvijD.C. "the ball does negative work on you" No. If a system increases in KE, positive work was done on it. The only thing doing work on the body is the force from the ball. It's doing positive work because the direction of motion is in the same direction as the force. Negative work is done when the kinetic energy of a body decreases. Commented May 27, 2020 at 13:40
• @BillN No, the work being done on the body is not only from the ball. It's also from the muscles of the body. When you push the ball, the point of contact between you and the ball is moving in the opposite direction of the force being applied to you by the ball.
– user87745
Commented May 27, 2020 at 13:43
• @DvijD.C. I don’t think that the work done by the ball is negative. Negative work, by work-energy theorem, suggests a decrease in SPEED. Since the body clearly increased in SPEED, something did POSITIVE work on it.
– Lyle
Commented May 27, 2020 at 13:54

You did work on yourself by throwing the ball. Your body used some energy to do so as did a rocket by using its fuel.

• Can a system really do work on the same system?
– Lyle
Commented May 26, 2020 at 2:17
• Think about it this way if you want to convince yourself. When you swim, you move forward, and the only work done is by you and you alone Commented May 26, 2020 at 8:02