## Question 2:

### Part (a):

| Working/Answer | Mark | Guidance |
|---|---|---|
| Using impulse-momentum principle for $B$ | M1 | Correct no. of terms, dimensionally correct, condone sign errors, must be difference of momenta |
| $5mu = 3m(v_B - {-u})$ | A1 | Correct unsimplified equation |
| $v_B = \frac{2u}{3}$ | A1 | Correct appropriate velocity |
| Use of conservation of momentum | M1 | Use of CLM with correct no. of terms, dimensionally correct, condone sign errors. **Alternative:** Use impulse-momentum for $A$ |
| $4mu - 3mu = 2mv_A + 3mv_B \left(= 2mv_A + 3m \cdot \frac{2u}{3}\right)$ | A1ft | Correct unsimplified CLM equation. **Or:** $-5mu = 2m(v_A - 2u)$ |
| $v_A = -\frac{u}{2}$ | A1 | Correct speed |
| Use of NLR | M1 | Use of NLR with $e$ on correct side |
| $e = \frac{v_B - v_A}{2u + u} = \left(\frac{\frac{u}{2} + \frac{2u}{3}}{2u + u}\right)$ | A1ft | Correct unsimplified equation |
| $e = \frac{7}{18} = 0.39$ or better | A1 | Correct answer |

**(9 marks)**

### Part (b):

| Working/Answer | Mark | Guidance |
|---|---|---|
| KE Loss $=$ Initial KE $-$ Final KE | M1 | Correct no. of terms, must be a difference. Dimensionally correct at point of stating expression for loss/change in KE |
| $= \frac{1}{2} \cdot 2m(2u)^2 + \frac{1}{2} \cdot 3mu^2 - \left(\frac{1}{2} \cdot 2m\left(-\frac{u}{2}\right)^2 + \frac{1}{2} \cdot 3m\left(\frac{2u}{3}\right)^2\right)$ | A1ft | Unsimplified expression in $u$ with at most 1 error, ft on speeds from (a) |
| | A1ft | Correct unsimplified expression in $u$, ft on speeds from (a) |
| $= \frac{55mu^2}{12}$ | A1 | Accept $4.58mu^2$ or $4.6mu^2$ |

**(4 marks)**

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