## Question 7(a):

Equation for conservation of energy | M1 | Need all terms. Condone sign errors

$\frac{1}{2} \times m \times 144 + m \times g \times 20 = \frac{1}{2}mv^2$ | A1 | Correct unsimplified equation with at most one error

| A1 | Correct equation (with or without $m$)

$v = 23$ or $23.2$ | A1 | Max 3 s.f.

**[4]**

## Question 7(b):

$12\cos\theta \times 5 = 40$ | M1 | Horizontal motion. Condone sine/cosine confusion

(minimum$=$) $12\cos\theta = 8\ (\text{m s}^{-1})$ | A1 | Final answer: do not ignore subsequent working

**[2]**

## Question 7(c):

Speed $= 10 \Rightarrow$ Vertical component $= 6\ (\text{m s}^{-1})$ | B1ft | Follow their horizontal component

$(\pm)6 = 12\sin\theta - gt$ | M1 | Vertical speed

$= 12 \times \frac{\sqrt{5}}{3} - gt$ | A1 | Correct equation for one value of $t$ or for the time interval. Correct trig value seen or implied

$(t = 0.30..$ and $t = 1.52..)$

Time $= 1.52.. - 0.30.. = 1.22...\ \text{(s)}$ | A1 | Correct interval

Required time $= 5 - 1.22\ \text{(s)}$ | M1 | Find required time – follow their $1.22$

$= 3.78\ \text{(s)}$ | A1 | Or $3.8$. Max 3 s.f.

**[6]**

**Alternatives for M1A1A1:**

Use of $v = u + at$: $-6 = 6 - gt$ | (M1) | Or find time to top and double it

$t = \frac{12}{g}$ | (A1) |

Vertical speed: $6 = 12\sin\theta - gt_1$ | (M1) |

$-6 = 12\sin\theta - gt_2$ | (A1) |

$12 = g(t_2 - t_1),\ t_2 - t_1 = \frac{12}{g}$ | (A1) |

**Alternatives for B1M1A1A1:**

Height above $A$: $\frac{22}{g}$ | (B1) | Using energy. $2.24...$ seen or implied e.g. by $22.24...$

Use of $s = ut + \frac{1}{2}at^2$ | (M1) | $20 + \frac{22}{g}$ used with $12\sin\theta$ is M0

$\frac{22}{g} = 12\sin\theta\, t - \frac{1}{2}gt^2$ | (A1) |

Time $= 1.52.. - 0.30.. = 1.22...\ \text{(s)}$ | (A1) | Correct interval

Speed $10$, angle to horizontal: $\alpha \Rightarrow 10\cos\alpha = 8$ | (B1) |

Time to top: $0 = 10\sin\alpha - gt,\ 10 \times 0.6 = gt$ | (M1)(A1) |

Total time $= \frac{12}{g}$ | (A1) |

**[11]**

## Question 8a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Impulse on $A$: $8mu = 3mv - 3m \times \frac{u}{3}$ | M1 | Terms dimensionally correct. Must be subtracting. Condone sign errors. Must be combining correct mass and speed |
| $v = 3u$ | A1 | |
| Impulse on $B$: $8mu = 4mu + 4mw$ | M1 | Terms dimensionally correct. Condone sign errors. Or use CLM: $9mu - 4mw = 3m\frac{u}{3} + 4mu$. Must be combining correct mass and speed |
| $w = u$ | A1 | |
| Impact law: $u - \frac{u}{3} = e(3u + u)$ | M1 | Used the right way round. Condone sign errors |
| $e = \frac{1}{6}$ | A1 | |
| **Total** | **[6]** | Award first 4 marks in order. Watch out for sign errors. If $3mv + 4mw$ in CLM combined with $w = -u$, maximum score is 4/6 |

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## Question 8b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Gap when $B$ hits wall $= \frac{2d}{3}$ | B1 | Or find distances from first impact: $s_A = \frac{d}{3} + \frac{u}{3}t$ and $s_B = d - \frac{u}{4}t$ |
| Speed of rebound from wall $= \frac{u}{4}$ | B1 | Allow $+/-$ |
| Time to close gap $= \dfrac{\frac{2d}{3}}{\frac{u}{3} + \frac{u}{4}}$ | M1 | |
| $= \frac{8d}{7u}$ | A1 | |
| Distance from wall $= \frac{8d}{7u} \times \frac{u}{4}$ | DM1 | Dependent on the preceding M1 |
| $= \frac{2d}{7}$ | A1 | |
| **Total** | **[6]** | |

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## Question 8balt:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Time for $A$: $\frac{d-x}{\frac{u}{3}} \left(= \frac{3d-3x}{u}\right)$ | B1 | |
| Speed of rebound from wall $= \frac{u}{4}$ | B1 | |
| Time for $B$: $\frac{d}{u} + \frac{x}{\frac{u}{4}}$ | M1 | |
| $= \frac{d + 4x}{u}$ | A1 | |
| $3d - 3x = d + 4x$ | DM1 | Solve for $x$. Dependent on preceding M1 |
| $x = \frac{2d}{7}$ | A1 | |
| **Total** | **[6]** | |
| **Overall Total** | **[12]** | |