Need help with question and checking of answers

[rock.freak667]

Homework Statement

http://img234.imageshack.us/img234/9536/scan0006hj0.jpg
Two identical snooker balls are at rest at the point A and B of a snooker table, where AB+1m. The liunes AB is aparallel to a smooth side of the table and at a distance of 0.4m from that side. The ball A is projected along the table with a speed of 2ms^{-1} to hit the side of the table at C. It rebounds and subsequently hits the ball at B. The angles that AC and CB make with the table are \alpha and \beta respectively (see diagram). The coefficient of resititution between the side and a ball is \frac{1}{4}. Resistances may be ignored and the balls may be considered as particles.

(i)Show that cot\alpha + cot\beta=2.5 and tan\alpha=2
(ii) Show that the direction of motion of the ball turns through 90 degrees on impact at C
(iii) find the speed of the ball after impact at C.

Homework Equations

e=\frac{\delta v}{\delta u}

The Attempt at a Solution

Well here is my work...in pictures because, typing it up would have taken a long time.

http://img233.imageshack.us/img233/2581/scan0006cb8.jpg

http://img396.imageshack.us/img396/4219/scan0007wu2.jpg


Strangely enough,I had a hard time looking for ways to show that tan\alpha=2

 

[Doc Al]

Strangely enough,I had a hard time looking for ways to show that tan\alpha=2

One thing that will help with both showing that \tan\alpha = 2 and finding the speed after impact is realizing that the coefficient of restitution only applies to the component of velocity perpendicular to the surface.

 

[rock.freak667]

So the perpendicular comp. of the velocity before impact is 2sin\alpha

and the perpendicular comp. of velocity after is vsin\beta

and

\frac{1}{4}=\frac{vsin\beta}{2sin\alpha}


But where would I get the tan from?

 

[Doc Al]

But where would I get the tan from?

Use both the horizontal and vertical components.

 

[rock.freak667]

Use both the horizontal and vertical components.

2cos\alpha and vcos\alpha, both point in the same direction.

So 2cos\alpha=vcos\beta because the horizontal velocity should stay the same?

 

[Doc Al]

Yes. Now relate the tangents by dividing the vertical and horizontal components.

 

[rock.freak667]

ah thanks, got it there.