But as is observed in the question, there are many times where momentum is apparently _not_ conserved. I can name two more obvious cases, with less "complication" (i.e. rolling): A ball drops from a height, and gains momentum. A ball of putty strikes a wall, sticks and stops. These all apparently violate conservation of momentum.
However, these problems are all apparent. Momentum is conserved only for a closed system. In all these cases, there are external forces. But, we should also be able to find where the momentum went. It turns out, in all these cases, the body which needs to be included to let momentum be conserved is the earth. When a body drops from a height, it gains momentum down, while the earth gains the same momentum up. Since the earth is very massive, you can not observe its motion in reaction. Same goes for a ball rolling downhill. And, in the case of putty ball hitting the wall and stopping, the momentum is passed onto the earth, which is again, not observable.
Answered by: Yasar Safkan, Ph.D., Software Engineer, GVZ., Istanbul, Turkey
Take the following example:
A 10 kg ball is dropped from rest and allowed to fall for one second. The force of gravity pulling down on the ball is 98 N, causing an acceleration of a=F/m = 98N/ 10kg = 9.8m/s2. Thus after one second the ball is traveling downwards at a velocity of 9.8m/s and its momentum p=mv = 98 kg m/s.
At the same time the Earth is pulled up with a force of 98N. Its mass is about 6 x 10^24 kg so its acceleration is a tiny 98 N/ 6 x 1024 kg = 1.6 x 10-23 m/s2. Thus after the 1 second the Earth will be moving upwards with a velocity of 1.6 x 10-23 m/s. To find the Earth's momentum we multiply its mass times its velocity and get... 98 kg m/s upwards!
Answered by: Rob Landolfi, Science Teacher, Washington, DC
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