How Feynman proves momentum is conserved in this example?

Question

Here is what Feynman says in section I.10-3:

"Suppose we have two equal masses, one moving with velocity $v$ and the other standing still, and they collide and stick; what is going to happen? There is a mass $2m$ altogether when we are finished, drifting with an unknown velocity. What velocity? That is the problem. To find the answer, we make the assumption that if we ride along in a car, physics will look the same as if we are standing still. We start with the knowledge that two equal masses, moving in opposite directions with equal speeds v, will stop dead when they collide. Now suppose that while this happens, we are riding by in an automobile, at a velocity -v. Then what does it look like? Since we are riding along with one of the two masses which are coming together, that one appears to us to have zero velocity. The other mass, however, going the other way with velocity v, will appear to be coming toward us at a velocity 2v. Finally, the combined masses after collision will seem to be passing by with velocity v. We therefore conclude that an object with velocity 2v, hitting an equal one at rest, will end up with velocity v, or what is mathematically exactly the same, an object with velocity v hitting and sticking to one at rest will produce an object moving with velocity v/2."

Feynman said,

"two equal masses, moving in opposite directions with equal speeds, will stop dead when they collide."

If so, then the car we are riding, moving with velocity -v, and the other mass, moving with velocity v, collide, they must stop dead as their masses are same. Then why do they keep moving with half of the velocity?

Answer 1

... the car we are riding, moving with velocity $-v$, and the other mass, moving with velocity $v$, collide, they must stop dead as their masses are same.

The car is not involved in the collision - it simply provides a moving platform from which we can observe the two objects that are colliding. Feynman uses the car to introduce the idea of a moving inertial frame of reference without actually calling it that.

After the objects collide the combined object is stationary relative to the ground. However, relative to the car (which is still moving with velocity $-v$) the combined object is moving with velocity $v$ after the collision. So from the point of view of the car the velocities of the objects before the collision were $2v$ and $0$, and after the collision the combined object has a velocity of $v$. By halving the velocities, Feynman concludes that if an object moving with velocity $v$ hits an identical stationary object, the combined object will have velocity $\frac v 2$ after the collision.