Conservation of Kinetic Energy Proof

[Davey Boy]

[EDIT]Title is incorrect, it is a proof using Conservation of Energy/Momentum

Homework Statement

You take a pool shot for the win. The perfectly [EDIT]elastic[/EDIT] collision is such that after the collision the Que ball has a non-zero velocity along angle $$\theta$$, and the 8 ball has a non-zero velocity along angle $$\beta$$. Prove that $$\theta$$+$$\beta$$=90$$\circ$$. The two pool balls have exactly the same mass.

Homework Equations

No equations are given, but I am fairly certain only $$\Delta$$K=$$\Delta$$P=0 is needed. My teacher vaguely hinted that those were the equations/principals he recommend we utilize.

The Attempt at a Solution

I came up with 2 equations:
1) [1/2]M*V_IQ²=[1/2]M*V_FQ²*cos($$\theta)$$+[1/2]M*V_F8²*cos($$\beta$$)
2) M*V_IQ=M*V_IQ*cos($$\theta)$$+M*V_F8cos($$\beta$$)
I solved for V_FQ and plugged that into the 1^st equation. After simplifying I got;
2[(V_F8cos($$\beta$$))(V_FQcos($$\theta$$))]=0
I figured out that this means one of those variables must be 0.I know that V_FQ is not, and V_F8 is not, therefore, one of the angles must be equal to 90$$\circ$$ for that equation to equal 0.
I believe my initial equations are wrong, though i could have easily made an algebraic mistake. Any pointers would be very nice, Thanks!

 

[cepheid]

Hi Davey Boy, welcome to PF.

Are you sure that the problem statement shouldn't say "the perfectly elastic collision?" If not, then why are you using conservation of kinetic energy?

Your first equation, (the energy equation), is wrong. Why do you have those cosine factors in there?

 

[Davey Boy]

Hi Davey Boy, welcome to PF.

Are you sure that the problem statement shouldn't say "the perfectly elastic collision?" If not, then why are you using conservation of kinetic energy?

Your first equation, (the energy equation), is wrong. Why do you have those cosine factors in there?

It should be elastic rather than inelastic, my mistake. I'll edit that part. I have the cosine factors to account for the angle at which the pool balls are traveling after the collision. I know those are needed for the momentum equation because my teacher did a brief example for 2-D momentum problems, but are they not for kinetic energy?

 

[cepheid]

It should be elastic rather than inelastic, my mistake. I'll edit that part. I have the cosine factors to account for the angle at which the pool balls are traveling after the collision. I know those are needed for the momentum equation because my teacher did a brief example for 2-D momentum problems, but are they not for kinetic energy?

Energy doesn't have a direction. The kinetic energy of a ball ( its energy due to motion) depends upon its *speed*, i.e. the magnitude of its velocity. So KE = (1/2)mv², period. The energies of the two balls after the collision must add up to the energy of the que ball before.

 

[Davey Boy]

Energy doesn't have a direction. The kinetic energy of a ball ( its energy due to motion) depends upon its *speed*, i.e. the magnitude of its velocity. So KE = (1/2)mv², period. The energies of the two balls after the collision must add up to the energy of the que ball before.

Awesome, that's what I had suspected. Thanks for the help, I'll try that out and get back to you if it works or not.

 

[Davey Boy]

Thanks to the hint of energy not being a vector, thus not requiring a direction, I was able to find equations for V_FQ and V_F8. I have no idea what to do now though. The equations I got are:
V_FQ=$$V_{IQ}(cos\beta-1)/(cos\beta-cos\theta)$$
and
V_F8=$$V_{IQ}(1-cos\theta)/(cos\beta-cos\theta)$$

 

[cepheid]

Hey, well I noticed that your momentum conservation equation was only for conservation of momentum in one direction. There's also conservation of momentum in the other, perpendicular direction (which gives you another equation to play with). That equation will relate VF8 to VFQ.