Elastic Billiard Ball Collision

[iandelaney]

Homework Statement

A cue ball with speed U hits a stationary red ball of equal mass. The collision is elastic (ie no energy is converted into other forms). After the collision the cue ball is moving at an angle θ to its original path. Find the final speed of the cue ball.

Homework Equations

conservation of momentum: mu=mv₁mv₂

conservation of energy: 1/2mu² = 1/2mv₁² + 1/2mv₂²

The Attempt at a Solution

As I understand it this should be a fairly standard problem but I have been struggling to find any help on the tinterweb hence my posting :-)

I know that from the conservation of momentum that the velocities of the two balls after the collision will be equal to the velocity of the cue ball initially. I would then like to substitute u=v₁+v₂ into the conservation of energy to obtain (v₁+v₂)² = v₁²+v₂² which is only the case when v₁ and v₂ are zero. I was wondering if anyone could spot my mistake?

thanks in advance
Ian

 

[clamtrox]

The velocities are vectors, remember that. You can get rid of the velocity of the red ball using conservation momentum, plug that into the equation of conservation of energy. You get the angle dependence from the dot product between the two velocities.

 

[iandelaney]

When I do that I obviously find that v₂=u-v₁ and when I sub this into the conservation of energy equation it still comes out as either v₁=u or v₁=0, and I am assuming that this can't be the case as it states in the question that the ball continues to move after the collision at an angle θ :s

 

[Ray Vickson]

Homework Statement

A cue ball with speed U hits a stationary red ball of equal mass. The collision is elastic (ie no energy is converted into other forms). After the collision the cue ball is moving at an angle θ to its original path. Find the final speed of the cue ball.

Homework Equations

conservation of momentum: mu=mv₁mv₂

conservation of energy: 1/2mu² = 1/2mv₁² + 1/2mv₂²

The Attempt at a Solution

As I understand it this should be a fairly standard problem but I have been struggling to find any help on the tinterweb hence my posting :-)

I know that from the conservation of momentum that the velocities of the two balls after the collision will be equal to the velocity of the cue ball initially. I would then like to substitute u=v₁+v₂ into the conservation of energy to obtain (v₁+v₂)² = v₁²+v₂² which is only the case when v₁ and v₂ are zero. I was wondering if anyone could spot my mistake?

thanks in advance
Ian

Such problems are almost always easier to solve if you first go into the CM (center-of-mass) system, then transform back to the lab system to finally finish the calculation. In the CM system, the total momentum = 0 before (and after) the collision, so if the masses of the balls are equal, their speeds are equal in the CM system before and after the collision. If energy is conserved in the lab system it is also conserved in the CM system, and vice-versa.