Mass in conservation of linear momentum Imagine I throw two objects at each other.

*

*One object is a small rigid body and the other is a very long rod.

*The impact will happen at the closest sections of the two objects,
assumed to be normal to their respective longitudinal axis and parallel to each other.

*The line of impact is contained in a plane that also contains the principal axis of inertia of both objects.

*The mass of the rigid body is $\ M_{S}$ (kg) and just before hitting the end section of
the rod, it has a known velocity $\ v_{Sb}$ (aligned with the longitudinal axis of the rod).

*After hitting the rigid body has a known velocity $\ v_{Sa}$, in the opposite direction of $\ v_{Sb}$.

*The rod has a mass-length density equal to $\ m_{R}$(kg/m) and just before hitting the end section of
the rigid object, it has a known velocity $\ v_{Rb}$ (aligned with the longitudinal axis of the rod).

I'd like to find the final velocity, $\ v_{Ra}$, of the rod after impact.
From the conservation of linear momentum:
$$\ v_{Sb}*M_{S}-v_{Rb}*M_{Rb}=v_{Ra}*M_{Ra}-v_{Sa}*M_{S}$$
But how to define $\ M_{R}$?
Before collapse it can be assumed that the entire rod is traveling with speed $\ v_{Rb}$, so $\ M_{Rb}$is the total mass of the rod.
But immediately after impact?
I know that the impact wave will propagate through the rod at the speed of sound through the material of the rod. Therefore, how to define $\ M_{Ra}$?
For a given time instant $\ t$ do I need to do the integral of the momentum of the rod along its length with a speed diagram? If yes, which speed diagram?
 A: You can get a rough estimate if you consider that at the moment of impact interacting masses are 0, and as time moves forward, larger part of the rod is getting involved in collision process. When shockwave covered half of the rod, half of the rod is participating in collision. When shockwave has covered the entire rod - entire rod is participatin in collision.
In case of a rod interacting mass will grow linearly from 0 to rod's mass, in time from collision from 0 to time shockwave needs to traverse the entire rod.
Better still to take into account that returning wave matters, and take speed of sound as 0.5 times of the given value, or take distance that it needs to cover as 2 times.
Lets consider ball mass 100, rod mass 100, ball momentum 100, rod momentum 0.
Before the collision rod speed is 0, after the collision rod speed is 0.5. To check if change is linear we need to check some more points.
In the middle of collision, assuming inelastic collision and ball is connected to the rod, far part of the rod speed is 0. Close part and ball, their mass together is 100+100/2, and momentum available is 100, so the speed is 100/150 = 0.67. On average rod's speed is 0.33.
So we have this progression: 0, 0.33, 0.5
Doesnt seem linear. I dont know this sequence.
