# Question 1:

## Part (a)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Diagram: before $u$ ms$^{-1}$ and $\frac{u}{3}$ ms$^{-1}$; after $\frac{u}{2}$ ms$^{-1}$ and $V$ ms$^{-1}$; P=5kg, Q=30kg | B1 | Accept $V$ in either direction. Given velocities and masses must be correct |
| PCLM: $5u - 30\frac{u}{3} = -5\frac{u}{2} - 30V$ | M1 | PCLM. Allow sign errors only |
| $V = \frac{u}{12}$ | A1 | Award even if direction of $V$ used in PCLM does not match diagram, so $\frac{u}{12}$ or $-\frac{u}{12}$ will get this A1 |
| Direction of $V$ correct | A1 | WWW. Direction of $V$ correct (may be implied from diagram) |
| $e = \dfrac{\frac{u}{2} - \frac{u}{12}}{u + \frac{u}{3}}$ | M1 | FT their $V$: allow sign errors, but must be right way up |
| $= \dfrac{\frac{5}{12}}{\frac{4}{3}} = \frac{5}{16}$ (= 0.3125) | A1 | cao |
| **[6]** | | |

## Part (a)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\rightarrow +ve$ | | |
| $30\left(-\frac{u}{12} - \left(-\frac{u}{3}\right)\right)$ | M1 | Allow sign errors |
| $7.5u$ | A1 | |
| Or: find impulse on P **and** reverse the sign | M1 A1 | $5\left(-\frac{u}{2} - u\right) = -\frac{15}{2}u$ **and** $7.5u$ cao. M0A0 unless sign is reversed |
| Direction must be given (may be implicit from diagram) | | |
| **[2]** | | |

## Part (b)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| As the parts move at 90°, PCLM in final directions | M1 | |
| For 2kg: $5 \times 6\sin\alpha = 2u$ | M1 | PCLM |
| $5 \times 6 \times \frac{3}{5} = 2u$ | A1 | Any form |
| $u = 9$ | A1 | |
| For 3kg: $5 \times 6\sin\beta = 3v$ | M1 | PCLM |
| $5 \times 6 \times \frac{4}{5} = 3v$ | A1 | Any form |
| $v = 8$ | A1 | |
| **Or** PCLM $\rightarrow$ $5 \times 6 = 2u\sin\alpha + 3v\sin\beta$ | M1 | PCLM. Allow cos instead of sin if error in both terms; allow sign errors; masses need to be there. Award if embedded in vector method |
| $30 = 2u \times \frac{3}{5} + 3v \times \frac{4}{5}$ | A1 | Any form |
| PCLM $\uparrow$: $2u\cos\alpha - 3v\cos\beta = 0$ | M1 | PCLM. Allow sin instead of cos if error in both terms and cos used in previous PCLM eqn; allow sign errors; masses need to be there. Award if embedded in vector method |
| $2u \times \frac{4}{5} = 3v \times \frac{3}{5}$ | A1 | Any form |
| Solving | M1 | A complete method involving 2 equations each in $u$ and $v$ |
| $u = 9$ | A1 | cao for one of $u$ or $v$ |
| $v = 8$ | F1 | for the other: FT substitution into their eqn |
| **[7]** | | Note: Award SC5 for $v=6$, $u=12$ (from cos/sin reversal). Uses velocity instead of mmtum: M0M0M1A0F1 max 2/7. Uses mass in one eqn only: M1A1M0M1A0F1 max 4/7 |

## Part (b)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Delta KE = \frac{1}{2}\times 2\times 9^2 + \frac{1}{2}\times 3\times 8^2 - \frac{1}{2}\times 5\times 6^2$ | M1 | M1 for attempt at difference of KE (3 terms of correct form) |
| $= 87$ J | A1 | cao |
| **[2]** | | |

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