Question

Question #69f5e

Answer 1

It is stated for 2-objects colliding in an isolated system:
The total momentum before and after the collision is equal.

Explanation:

Newton's `color(red)"3rd-law"`: If an object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A.

By Newton's `color(blue) "2nd-law"`, this force is equal to the product of the mass and the acceleration of the objects, so the product of the mass and acceleration of object A is equal but opposite to the product between the mass and acceleration of object B.

`(m a)_A = (m a)_B`

Acceleration is the change of velocity divided by the time. Time is cancelled in the equation because in a collision, the forces act within the same time frame. Thus, the product of object A's mass and change in velocity is equal but opposite to the product of object B's mass and change in velocity. `color(red)"Momentum"` is defined as the product of mass and velocity, so finally the change in object A's momentum is equal to the opposite of the change in object B's momentum.

`(m v)_A = (m v)_B`

NOTE: In a non-isolated system, an external force is applied during the collision, changing the original equation and causing momentum to differ before and after the collision.