Confusing linear momentum GRE question

[PsychonautQQ]

Homework Statement

http://grephysics.net/ans/8677/44

Homework Equations

p=mv

The Attempt at a Solution

Can somebody show me the math on why this is true? I feel like in my head if a particle hits the stick in the center of mass instead of at the end it would then make the center of mass of the stick move at greater velocity. This problem is showing that no matter where the particle hits the stick the center of mass of the stick has the same velocity and angular momentum doesn't matter? I'm confused ;-(

 

[TSny]

In order for the particle to come to rest after making an elastic collision with the end of the rod, the ratio of the masses m/M must be a certain value (which you can work out as an exercise). If you let the same particle strike the center of the stick, then you cannot assume that the particle will now come to rest. You can work out the final speeds of the particle and the stick and see if your intuition is right about the center of the stick having a greater final speed when it gets hit at the center.

 

[prasa043]

Thanks, TSny. I was working out some equations with conservation of energy in this case and I was wondering why I had a possible imaginary value for angular velocity. This now makes sense.

 

[PsychonautQQ]

In order for the particle to come to rest after making an elastic collision with the end of the rod, the ratio of the masses m/M must be a certain value (which you can work out as an exercise). If you let the same particle strike the center of the stick, then you cannot assume that the particle will now come to rest. You can work out the final speeds of the particle and the stick and see if your intuition is right about the center of the stick having a greater final speed when it gets hit at the center.

Am i correct in my conclusion that if the particle hits the middle of the stick the sticks center of mass will have the same velocity as if it hit the same but the particle will retain some of it's velocity? If the particle hits the end of the stick it's transfers momentum into the stick in linear and angular form but if it hits the center of mass then it only transfers linear momentum? and the linear momentum transferred to the stick is independent of where the particle collides with it?

 

[D H]

If the particle hits the stationary stick, after which the particle is stationary and the stick is moving, the velocity of the stick's center of mass will be $\frac m M v$ regardless of where the particle hits.

[strike]I do have an issue with that answer. It's the "One could use energy, but then one would have to take into account the inertia." There's nothing in the question that says the collision is elastic. You can't use conservation of (mechanical) energy if mechanical energy isn't conserved.[/strike]

Issue retracted. The question specifically says the collision is elastic.

 

[TSny]

Am i correct in my conclusion that if the particle hits the middle of the stick the sticks center of mass will have the same velocity as if it hit the same but the particle will retain some of it's velocity?

I'm not sure what you mean by "as if it hit the same...".

If the ratio of the masses is such that the particle comes to rest when it strikes one of the ends of the stick, then it will not come to rest when it strikes any of the other points of the stick. For these other points, the particle will have a velocity after the collision and you should be able to determine if the final velocity of the particle is in the same direction of the initial velocity or opposite direction ("bounces back").

If the particle hits the end of the stick it's transfers momentum into the stick in linear and angular form but if it hits the center of mass then it only transfers linear momentum?

Yes.

and the linear momentum transferred to the stick is independent of where the particle collides with it?

No, the change in linear momentum of the stick will depend on where the stick is struck by the particle.