Question

An elastic collision between two particles,one of mass M and the other of mass 4M.
Which particle feels a greater force?
Justify your answer mathematically

Answer 1

They both will feel the same amount of force. Basically, this is suggested by Newton's third law of motion, forces appears in pairs: every action has equal and opposite reaction. For example: If particle 1 applies force on particle 2, then particle 2 will also apply the same magnitude of force (but in opposite direction) on particle 1.

The laws are assumptions (axioms) that cannot be proved mathematically and they are established through empirical (experimental) experiences. However, laws can be proved mathematically when more fundamental assumptions are made, assumptions that have more universal nature than the ones to be proved.

Now, if we consider conservation of momentum as a more fundamental law (in fact, it is ! It holds even for collisions of fundamental quantum particles like electrons, quarks, etc. and the third law does not hold in Electrodynamics, but conservation of momentum holds in Electrodynamics), we can prove the third law.

Let particle 1 of mass and initial velocity , collides with a particle 2 of mass and velocity . The velocity of particle 1 after the collision is , and of particle 2 is .

Initial momentum of particle 1 is

Final momentum of particle 1 is

Initial momentum of particle 2 is

Final momentum of particle 2 is

Total initial momentum of the system is

Total final momentum of the system is

By the conservation of momentum

Change in Momentums of the two particles are related by

If the two particles were in contact for small time , the average force felt by particle 1 is

The average force felt by particle 2 is

Using the relation of change in Momentums of Particle 1 and Particle 2

We get (Newton's third law)

The magnitudes (our desired result)

Note that we had chosen masses of the colliding particles to be arbitrary, even if we substitute and our result will not change. The two particles will feel the same magnitude of force.

Also, note that we never made any assumption about the nature of collision. The result hold for both elastic and inelastic collisions.