Physics pool ball question, common-sense question?

Question

Okay, you hit the little white pool ball. The little white ball hits another ball (lets say the 8 ball).

I think the 8 ball CAN NOT travel at a faster speed than the white ball you initially hit (given a flat table, no wind, etc). My friend is arguing that it can. Can someone give me any pertinent physics or equations to back me up on this?

Answer 1

actually the cue ball is smaller (at least on standard UK tables, i had a summer job once in a pub with 2 tables)

the equations are in here
http://en.wikipedia.org/wiki/Elastic_collision

i'm trying to prove it to myself...

you could both be right/wrong. it depends on the balls

here m1 is mass of cue ball, u1 is speed of cueball before collision, v1 is speed of it after
m2 u2 and v2 are as before for 8ball
3 cases:
~~~
if the 2 are equal,
the momentum is simply transferred:
looking at the 3rd equation in the link, u2 is 0 (8ball not moving) it becomes
v1= u1(m1-m2)/(m1+m2)
and
v2=2m1u1/(m1+m2)

if m1=m2 the result for v1 is 0 (m1-m2=0)

~~~
if m1
rearrange v2 eqn to
(1+m2/m1)(v2)=2u1

since m2>m1 the tern m2/m1 is a little above 1
so you get 1+(a little over 1)(v2)= (a little over 2)v2 =2u1
v2 = a little less than u1

and v1 eqn m1 - m2 is slightly negative (-ve)
v1 = u1(small -ve)/ something +ve

so the cue ball is going backwards towards the point you hit it. (like hitting a ball against the wall, it goes back. the wall has more mass!)

~~~
m1>m2

rearrange v2 eqn to
(1+m2/m1)(v2)=2u1

m2/m1 is less than 1, so (1+ almost 1)v2=2u1
becomes
v2= 2u1/(almost 2)
2/almost 2 is greater than 1

so v2>u1 which makes your friend right in this case


to finish
for v1 eqn v1= u1(m1-m2)/(m1+m2)
m1-m2 is small but positive

so v1 is small: the cue ball rolls forward slowly

~~~
so you're both wrong!
it depends how big the balls are relative to each other!
BUT if you want to convince him you're right, check the size of the balls where you play pool... and you can just show him the relevant work :)

having said that, this is in a model. on a real table you have spins and stuff you can apply, and friction etc etc

while you're not happy with the result, i hope you're happy i spent my time doing that for you!

Answer 2

On tables at bars, the cue ball fall out of the table after a scratch and the regular balls do not. This is because the cue ball is heavier or slightly larger diameter or both. If the cue ball has higher mass, conservation of momentum says the 8 can have higher velocity than the cue ball.

Answer 3

let the white ball hits the ball 8 with momentam 'P' .from law of moment conservation ball 8 will travel with 'P' so 8's speed may not grater then white ball.