M4 June 2007 Q1
1. A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of 60\(^\circ\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of 30\(^\circ\) with the wall.
| Scheme | Marks |
|---|---|
| \(u\cos 60^0 = v\cos 30^0\) | M1A1 |
| \(u = v\sqrt{3}\) | A1 |
| KE lost \(= \dfrac{1}{2}m\left(u^2 - v^2\right)\) | M1 |
| Fraction of KE lost \(= 1 - \left(\dfrac{v}{u}\right)^2\) | DM1 |
| \(= 1 - \dfrac{1}{3} = \dfrac{2}{3}\) or at least 3sf ending in 7 or \(\dfrac{3}{4}\left(1 - e^2\right)\) | A1 |
| (6) |
Notes
M1 Resolve parallel to the wall
Alt: reasonable attempt at equation connecting two variables
A1 Correct as above or equivalent
equation correct
A1 \(u\) in terms of \(v\) or v.v. – not necessarily simplified.
or ration of the two variables correct
M1 expression for KE lost
DM1 expression in one variable for fraction of KE lost – could be \(u/v\) as above
A1 cao
The first three marks can be awarded in (b) if not seen in (a)
| Scheme | Marks |
|---|---|
| \(e = \dfrac{v\sin 30^\circ}{u\sin 60^\circ}\) | M1A1 |
| \(= \dfrac{v}{u}\cdot\dfrac{1}{\sqrt{3}}\) | DM1 |
| \(= \dfrac{1}{3}\) | A1 |
| (4) | |
| (10 marks) |
Notes
M1 Use NIL perpendicular to the wall and form equation in \(e\)
A1 Correct unsimplified expression as above or \(eu\sin 60^o = v\sin 30^o\) or equivalent
DM1 Substitute values for trig functions or use relationship from (a) and rearrange to e = …..
A1 cao accept decimals to at least 3sf
The first two marks can be awarded in (a)