Elastic Collisions: billiard ball problem with a twist

[Benny851]

Homework Statement

A billiard ball ( mass = 10kg, initial velocity is 5 m/s) is launched along x-axis at a stationary billiard ball ( mass = 5kg). After collision, the first ball goes off at 30 degree angle above x-axis and 2nd ball goes off at 45 degree angle below x-axis. Calculate the final velocities of both billiard balls.

Homework Equations

M1V1 + M2V2 = M1V1final + M2V2final (conservation of momentum)

1/2M1V1^2 + 1/2M2V2^2 = 1/2M1V1final + 1/2M2V2^2

The Attempt at a Solution

Typically in most billiard ball problems you are not given both angle measurements, which means you need 3 equations: momentum in x direction, momentum in y direction and Kinetic energy. But, in this case I only have 2 unknowns, not 3. So my question is whether I need to even use the third kinetic energy equation? I don't understand how I can solve for 2 unknowns by using 3 equations. My gut tells me just to use the 2 momentum equations, which is what I have been doing, but whenever I read about elastic collision problems I see that KE equation is also used. Some guidance on this topic would be really appreciated. Thanks.

 

[gneill]

Momentum is always conserved, so if here is enough information given to solve the problem using only conservation of momentum, go for it!

 

[rrodriguez]

Your reasoning is correct, there's no need for the kinetic energy equations precisely because you know the angles they went off on. It's really just a way to test your understanding, to make sure you appreciate the underlying concept (conservation of linear momentum) by changing the usual form of the question. Mathematically, it's an example of what's sometimes called an overdetermined system (more equations than unknowns). Of course, once you know the final velocities you can then go ahead and calculate the kinetic energies, if you wished.

 

[Andrew Mason]

The initial kinetic energy is 125 J. When I work out the final speeds of the two balls using conservation of momentum, the total energy is a bit more than 125 J. So the directions given for the balls after the collision is not quite possible.

AM

 

[Nickie]

You are correct that in this problem, you only need to use the two equations for conservation of momentum in the x and y directions. The third equation for kinetic energy is not necessary because the problem does not require you to calculate the final velocities in terms of their kinetic energy. Instead, you can use the given information about the angles to find the components of the final velocities in the x and y directions. The conservation of momentum equations will allow you to solve for these components and then you can use trigonometry to find the magnitude and direction of the final velocities.