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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?

jpcgandre
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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.

Surprised Seagull
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  • Thanks for the answer. I agree. But in terms of speed profile, is it OK if I consider a linear variation of the speed after impact from $\ v_{Ra}$ at the section of impact to -$\ v_{Rb}$ at length x where x is the section of the rod where the front of the impact wave is at time instant t? – jpcgandre Feb 17 '22 at 17:19
  • And can I ignore the part of the rod which is still traveling at $ v_{Rb}$? – jpcgandre Feb 17 '22 at 17:27
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    @jpcgandre if rigid object is much heavier, or lighter, then change of speed of one of the bodies can be ignored. if object is similar mass - its change of speed is also important. If rod's matter accelerates instantly and only once as shockwave passes, early parts will accelerate more than the late parts, and rod wont have one speed. If rigid body connects to the rod on impact, inelastic, and speed of the rod's part that is connected to the rigid body and is covered by the shockwave is constantly synchronized, then yes, speed of the rod's two parts avg grows linearly and ends up all the same. – Surprised Seagull Feb 17 '22 at 17:38
  • Apologies, so to finish this off, I will be right if I ignore the part of the rod which is still traveling at vRb since this part's mass is not interacting with impact? – jpcgandre Feb 17 '22 at 17:57
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    @jpcgandre i was wrong. It appear to be not linear. Yes, far part of the rod cant interact before shockwave reaches it – Surprised Seagull Feb 17 '22 at 18:12
  • When you write "Close part and ball, their mass together is 100+100/2, and momentum available is 100, so the speed is 100/150 = 0.67", I think there's a contradiction since you also write that "after the collision rod speed is 0.5"... – jpcgandre Feb 17 '22 at 18:20
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    @jpcgandre I consider two cases: 1) collision is completed fully, no more interaction is needed. 2) in a middle of the collision, where shockwave has covered only half of the rod – Surprised Seagull Feb 17 '22 at 18:29
  • Thank you for the follow up – jpcgandre Feb 17 '22 at 18:30