Energy and momentum conservation laws problem

[Petrulis]

Homework Statement

Two balls go vis-a-vis (each of them speed is v) and strike. Hit is absolutely elastic. After the hit, one of the balls changes his motion direction 30 degrees. I need to find the direction of the other ball and both ball speeds after the hit.

The Attempt at a Solution

This problem is solved in center-of-mass frame of reference.

In this frame net momentum is zero, and total kinetic energy is conserved. Let Uo and Vo be the velocities of the balls before collision:

Vo = +v​

Uo = -v​

Energy KEo = 1/2 m (Vo² + Uo²) = mv²

After collison their velocities are V1 and U1. Since net momentum is conserved, and remains zero

V1 = -U1​

Energy after collison is

mv² = KEo = KE1 = 1/2 m (V1² + U2²) = mV1²
mv² = mV1²
Vo² = V1²​

Therefore, despite the velocity vector changed, its absolute value (i.e. speed) remains unchanged.

Answer:
both balls' absolute velocities did not change, and both their directions changed by 30 degrees.

Isn't here made any mistakes in the solution? And is it possible to solve the problem if the masses of the balls are different?

Thanks in advance.

 

[anathema]

Your solution is correct. In an elastic collision, the total kinetic energy and momentum are both conserved. In this case, since the total momentum is zero, the final velocities of the balls will be equal in magnitude but opposite in direction.

As for the masses of the balls, the solution will still hold true. The only difference is that the final velocities will be different for the two balls, depending on their masses. This can be solved using the equations for conservation of momentum and energy in a similar manner as you have done for equal masses.

Hope this helps!

 

[m2003]

I can confirm that there are no mistakes in the solution provided. The solution correctly uses the conservation laws of energy and momentum to solve the problem.

To solve the problem with different masses, the same principles can be applied but with the addition of the conservation of angular momentum. This would involve considering the rotational motion of the balls as well as their linear motion. The final velocities and directions of the balls would depend on the relative masses and initial velocities of the balls.