## Question 4:

### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of conservation of momentum | M1 | For use of CLM, with correct no. of terms, condone sign errors |
| $3mu - 2mu = 3mv + mw$ | A1 | For a correct equation |
| Use of NLR | M1 | For use of Newton's Law of Restitution, with $e$ on the correct side |
| $3ue = -v + w$ | A1 | For a correct equation |
| Using a correct strategy to solve, setting up two equations in $u$ and $v$ and solving for $v$ | M1 | For setting up two equations and solving for $v$ |
| $v = \frac{u}{4}(1 - 3e)$ | A1 | For a correct expression for $v$ |

### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{u}{4}(1 - 3e) < 0$ | M1 | For use of an appropriate inequality |
| $\frac{1}{3} < e \leq 1$ | A1 | For a complete range of values of $e$ |

### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solving for $w$ | M1 | For solving their equations for $w$ |
| $w = \frac{u}{4}(1 + 9e)$* | A1* | For the given answer |

### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $e = \frac{5}{9}$ | M1 | For substituting $e = \frac{5}{9}$ into their $v$ and $w$ |
| $v = -\frac{u}{6},\ w = \frac{3u}{2}$ | A1 | For correct expressions for $v$ and $w$ |
| Use NLR for impact with wall, $x = fw$ | M1 | For use of Newton's Law of Restitution, with $e$ on the correct side |
| Further collision if $x > -v$ | M1 | For use of appropriate inequality |
| $f\frac{3u}{2} > \frac{u}{6}$ | A1 | For a correct inequality |
| $1 \geq f > \frac{1}{9}$ | A1 | For a correct range |