$a = 5 - 2t \Rightarrow v = 5t - t^2 + 6$ | M1 A1, A1 |

$v = 0 \Rightarrow t^2 - 5t - 6 = 0$ | indep M1 |

$(t - 6)(t + 1) = 0$ | dep M1 |

$t = 6\text{ s}$ | A1 |

**Total: 6 marks**

## Question 2(a):

$\frac{P}{24} = 600 \text{ or } \frac{1000P}{24} = 600 \Rightarrow P = 14.4\text{ kW}$ | M1 A1 |

**Total: 2 marks**

## Question 2(b):

$\frac{30000}{20} - 1200 \times 9.8 \times \sin \alpha - 600 = 1200a$ | M1 A2, 1.0 |

$\Rightarrow a = 0.4\text{ m s}^{-2}$ | A1 |

**Total: 4 marks**

## Question 3(a):

$I = \pm 0.5(16\mathbf{i} + 20\mathbf{j} - (-30\mathbf{i}))$ | M1 |

$= \pm(23\mathbf{i} + 10\mathbf{j})$ | indep M1 |

$\text{magn} = \sqrt{23^2 + 10^2} \approx 25.1\text{ Ns}$ | indep M1 A1 |

**Total: 4 marks**

## Question 3(b):

$v = 16\mathbf{i} + (20 - 10t)\mathbf{j}$ | M1 |

$t = 3 \Rightarrow v = 16\mathbf{i} - 10\mathbf{j}$ | indep M1 |

$v = \sqrt{16^2 + 10^2} \approx 18.9\text{ m s}^{-1}$ | indep M1 A1 |

**Total: 4 marks**

## Question 4(a):

Total mass = $12m$ (used) | M1 |

(i) M(AB): $m.3a/2 + m.3a/2 + m.3a + 6m.3a + 2m.3a = 12m.x$ | indep M1 A1 |

$\Rightarrow x = \frac{5}{2}a$ | A1 |

(ii) M(AD): $m.a + m.a + m.2a + 6m.2a = 12m.y$ | indep M1 A1 |

$\Rightarrow y = \frac{4}{3}a$ | A1 |

**Total: 7 marks**

## Question 4(b):

$\tan \alpha = \frac{2a - 4a/3}{5a/2}$ | M1 A1 f.t. |

$\Rightarrow \alpha \approx 14.9°$ | A1 cao |

**Total: 3 marks**

## Question 5(a):

$x_A = 28t$ | $x_B = 35 \cos \alpha t$ | B1 B1 |

Meet $\Rightarrow 28t = 35 \cos \alpha t \Rightarrow \cos \alpha = 28/35 = 4/5$ * | M1 A1 |

**Total: 4 marks**

## Question 5(b):

$v_A = 73.5 - \frac{1}{2}gt^2$ | $v_B = 21t - \frac{1}{2}gt^2$ | B1 B1 |

Meet $\Rightarrow 73.5 = 21t \Rightarrow t = 3.5\text{ s}$ | M1 A1 |

**Total: 4 marks**

## Question 6(a):

M(A): $S.3a = 4mg.2a \cos \alpha + mg.4a \cos \alpha$ | M1 A1 |

$= \frac{48}{5}mga \Rightarrow S = \frac{16}{5}mg$ * | A1 |

**Total: 3 marks**

## Question 6(b):

R(↑): $R + S \cos\alpha = 5mg$ | M1 A1 |

R(→): $F = S \sin\alpha$ | M1 A1 |

$F \leq \mu R \Rightarrow \mu \geq \frac{48}{61}$ * | dep on both previous M's M1 A1 |

**Total: 6 marks**

## Question 6(c):

Direction of S is perpendicular to plank or No friction at the peg | B1 |

**Total: 1 mark**

## Question 7(a):

$R = 4g \cos \alpha = 16g/5 \Rightarrow F = 2/7 \times 16g/5$ | M1 A1 |

Work done $= F \times 2.5 = 22.4\text{ J}$ or $22\text{ J}$ | indep M1 A1 |

**Total: 4 marks**

## Question 7(b):

$\frac{1}{2} \times 4 \times u^2 = 22.4 + 4g \times 2.5 \times 3/5$ | M1 A2, 1.0 f.t. |

$\Rightarrow u \approx 6.37\text{ m s}^{-1}$ or $6.4\text{ m s}^{-1}$ | A1 cao |

**Total: 4 marks**

## Question 7(c):

$\frac{1}{2} \times 4 \times v^2 = \frac{1}{2} \times 4 \times u^2 - 44.8$ | M1 A2, 1.0 f.t. |

[OR $\frac{1}{2} \times 4 \times v^2 = 0 + 4g \times 2.5 \times 3/5 - 22.4$] |

$\Rightarrow v \approx 4.27\text{ m s}^{-1}$ or $4.3\text{ m s}^{-1}$ | A1 |

**Total: 4 marks**

## Question 8(a):

$mu = 4mw - mv$ | M1 A1 |

$eu = w + v$ | M1 A1 |

$\Rightarrow w = \left(\frac{1 + e}{5}\right)u, v = \left(\frac{4c - 1}{5}\right)u$ | indep M1 A1 A1 |

**Total: 7 marks**

## Question 8(b):

$w' = \left(\frac{4 + 4e}{25}\right)u$ | B1 f.t. |

Second collision $\Rightarrow w' > v$ |

$\Rightarrow \frac{4 + 4e}{25} > \frac{4e - 1}{5}$ | M1 |

$\Rightarrow e < 9/16$ | dep M1 A1 |

Also $v > 0 \Rightarrow e > 1/4$ Hence result (*) | B1 |

**Total: 5 marks**

## Question 8(c):

KE lost $= \frac{1}{2}mu^2 - \left[\frac{1}{2}.4m\left(\frac{(u/5)(1 + e)}{}\right)^2 + \frac{1}{2}m\left(\frac{(u/5)(4e - 1)}{}\right)^2\right]$ | M1 A1 f.t. |

$= \frac{3}{10}mu^2$ | A1 cao |

**Total: 3 marks**