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Suppose I have a system of a person holding a rock on a frictionless sheet of ice. He throws the rock at an angle above the horizontal to propel himself off the ice. Is the momentum of that system truly conserved?

If it will help you understand me, I'll post the assignment verbatim:

You are standing on a large, thick sheet of frictionless ice and holding a large rock. In order to get off the ice, you throw the rock so it has velocity 12.0 m/s relative to the earth at an angle of 35.0 degrees above the horizontal. Your mass is 70.0 kg and the rock's mass is 6.00 kg.

  1. Assume first the entire momentum of the person-rock system is conserved. What will be your speed and in what direction will you move? Is this a desirable outcome of your attempt to get off the ice?
    For the rest of the problem assume the ice does not break.
  2. Is truly the entire momentum conserved in this case? If not, which part of the momentum is not conserved? What is necessary to cause a change in momentum and what is the cause in this case?
  3. What will be your speed after you throw the rock?
  4. What is the kinetic energy of the system before and after you throw the rock? Is kinetic energy conserved? If not, where does the additional energy come from?
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    $\begingroup$ What's the difference between "conserved" and "truly conserved"? $\endgroup$
    – WillO
    Oct 7, 2015 at 2:53
  • $\begingroup$ Momentum is indeed conserved in the situation you presented. You should consider the (equal and opposite) momentum of the rock and the person throwing the rock. $\endgroup$
    – innisfree
    Oct 7, 2015 at 2:58
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    $\begingroup$ I suspect you are applying momentum conservation just to the rock or just to the person? One must of course consider the whole system. $\endgroup$
    – innisfree
    Oct 7, 2015 at 2:59
  • $\begingroup$ And the whole system is you+rock+earth, because you pushed on the earth when you threw the rock $\endgroup$
    – Philip Roe
    Apr 19, 2017 at 4:28

1 Answer 1

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Momentum is really always conserved. Truly. Throwing something upward (accelerating it with your arm) causes your feet to push harder on the ground. The increased down-force causes the ground under you, and ultimately the entire Earth, to shift direction downward.

Fortunately, the rock and the planet attract each other gravitationally, causing the rock to accelerate downward and the planet to accelerate upward. By the time they collide, the rock is coming down and the ground is going up.

Of course, the Earth is gigantic and your effect on its motion is minuscule. If you like, you can fairly say that the ground is the source (or sink) of endless momentum. Such a model is an approximation, not unlike saying the earth is flat. It's "correct enough" to give good answers to ordinary problems. Whether it's a good idea on homework, depends very much on the particular homework assignment.

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