Why the law of momentum conservation is not violated when a ball rolls down a hill and gains momentum?

Asked by:
Jackie

Answer

This happens to be a question I have pondered when I was studying laws of conservation. Considering mechanics only, conservation of energy is less lenient -- when you have two bodies colliding, you can lose some (or all) mechanical energy in the form of heat. So that is an easy explanation. However, it turns out conservation of momentum is stronger; independent of the amount of energy turned into heat or lost, momentum must be conserved...
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

We tend to think of the Earth (and other large bodies) as standing still for the purposes of balls rolling down hills etc. However as Newton's third law reminds us, every force has an equal and opposite force. Here, the gravitational force of the Earth on the ball causes the ball to accelerate, and thus gain momentum. However there is an equally strong force of the ball on the Earth, pulling the Earth up to meet the ball. The mass of the Earth is so large that the acceleration caused by the ball's gravity is usually negligible. However some quick calculations will show that the change in momentum of the Earth is equal in magnitude but opposite in direction to the change in momentum of the ball. Thus the net change in momentum is zero and the law of conservation of momentum is validated.
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/s^{2}. 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 10^{24} kg = 1.6 x 10^{-23} m/s^{2}. 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

'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'