Solving Momentum Problem With Perfectly Elastic Collisions

[Ed Quanta]

Two spheres are perfectly vertically alligned and falling towards the ground. They are separated by a small distance. The bigger sphere is on the bottom, and both spheres are supposed to have the common velocity v just before they strike the ground. I have to calculate the velocity of the smaller sphere, after the bigger sphere hits the ground. All collisions are perfectly elastic, and I am told to solve this as a sequence of closely spaced impulsive collisions. Help or ideas anybody? I am not sure what the sequence of collisions I am calculating are. And I am also not sure how to use either the lab or center of mass frame of references since the lab frame supposes that the velocity of one of the masses is zero which is not true, and the center of mass frame assumes the total momentum is zero, and I am not convinced that this is true.

 

[Integral]

The larger ball strikes the ground first (collision 1) it then collides with the smaller ball (collision 2). The larger ball will rebound with equal but opposite velocity. Use that velocity in the second collison.

 

[tomkeus]

Simply use the momentum and energy conservation laws. Thats enough to solve the problem.

 

[Ed Quanta]

Ok, but I am having trouble with what is going on in the second collision. I understand that when the larger mass hits the ground it will rebound with equal but opposite velocity. The smaller mass is going to collide with this mass with an equal but opposite velocity. However, since its mass is smaller, the momentum of the larger mass will be greater, and the total momentum will equal v(m-M) where m is the smaller mass, and M is the larger mass. However, this total momentum is non zero so I cannot use the center of mass frame?

-Mv + mv=v(m-M)=v1fM +v2fm

How am I to know the final velocities from this information alone? I want to know v2f, so v2f=(v(m-M)-v1fM)/m. But how am I to know v1f? I understand that momentum and energy are conserved, but how do the velocities change after the collision. It seems to me that they would be the same. By the way, v2f is supposed to equal v(3M-m/M+m).

 

[HallsofIvy]

Originally posted by Ed Quanta
Ok, but I am having trouble with what is going on in the second collision. I understand that when the larger mass hits the ground it will rebound with equal but opposite velocity. The smaller mass is going to collide with this mass with an equal but opposite velocity. However, since its mass is smaller, the momentum of the larger mass will be greater, and the total momentum will equal v(m-M) where m is the smaller mass, and M is the larger mass. However, this total momentum is non zero so I cannot use the center of mass frame?

-Mv + mv=v(m-M)=v1fM +v2fm

How am I to know the final velocities from this information alone? I want to know v2f, so v2f=(v(m-M)-v1fM)/m. But how am I to know v1f? I understand that momentum and energy are conserved, but how do the velocities change after the collision. It seems to me that they would be the same. By the way, v2f is supposed to equal v(3M-m/M+m).

You say "I understand that momentum and energy are conserved" but you have written only the momentum equation. As tomkeus suggested, use momentum AND ENERGY conservation.