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Consider a situation in which a body of mass m moving with a velocity v is collided with a similar mass, applying momentum conservation,the initial mass will come to rest and the other mass will move with a velocity of v.

  1. Velocity of first mass decreases while the other mass starts accelerating from rest.

  2. Both of them reach a common velocity of v/2. How could the mass in the front further accelerate to reach a velocity of v and the mass in back decelerate to zero velocity as there is no relative motion once they attain common velocity?

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    $\begingroup$ When the both velocities are equal, the bodies are still in contact and elastically deformed, so the force is not zero yet. $\endgroup$ Jan 27, 2021 at 17:11
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    $\begingroup$ Perhaps the animation in this previous answer of mine is of help physics.stackexchange.com/a/541440/93729. $\endgroup$ Jan 27, 2021 at 17:12
  • $\begingroup$ What you describe is exactly what happens in a completely inelastic collision. $\endgroup$
    – noah
    Jan 27, 2021 at 17:15

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In elementary dynamics we make two simplifying assumptions, which are linked:

  1. All bodies are completely rigid and do not deform.
  2. Momentum can be transferred instantaneously from one body to another (which implies infinite accelerations).

Both assumptions are unrealistic. In reality, all bodies deform to some extent, and transfer of momentum between colliding bodies takes a finite amount of time. As you say, gradual transfer of momentum means that the average velocities of colliding bodies are only equal at one instant in time, but deformation means that the bodies can stay in contact even though they have different average velocities.

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Momentum will be exchanged between the bodies (called an impulse). But typically this happens instantaneously for rigid bodies, so there is no in-between state. The total momentum is conserved if an amount $J$ is subtracted from one body and added to the other body.

But for deformable bodies, you can approximate the problem with a constant force $F$ acting on equal and opposite terms on each body resulting in the change in speed observed after some finite time $\Delta t$. The total momentum exchanged is $J = F \, \Delta t$, but since the moving body comes to rest, the impulse equals to $J=m v$, which is used to find the average force $$F = \frac{m v}{\Delta t}$$

You can represent the effect of this constant force graphically as the velocity of one body will steadily decrease, as the velocity of the other body will increase

fig1

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