## Question 5:

**Part 5a:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $P = Fv$ | M1 | $\frac{10000}{16}\ (= 625)$ seen or implied. Allow for $\frac{10}{16}$ |
| Equation of motion for the system | M1 | Dimensionally correct. Need all terms and no extras. Condone sign errors and sine/cosine confusion. If separate equations for van and trailer, just mark the combined equation |
| $F - 400 - 800g\sin\alpha = 800a$ | A1 | Unsimplified equation in $P$ or $F$ with at most one error |
| | A1 | Correct unsimplified equation in $P$ or $F$. Use of cosine in place of sine for both vehicles counts as a repeated error and only loses 1 mark |
| Obtain deceleration $0.419\left(\mathrm{ms}^{-2}\right)$ or $0.42\left(\mathrm{ms}^{-2}\right)$ | A1 | 3 sf or 2 sf only. Answer must be positive |
| **[5]** | | |

**Part 5b:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of motion for the van or the trailer | M1 | Dimensionally correct. Need all terms and no extras. Condone sign errors and sine/cosine confusion. Use the mass in the $ma$ term to decide which part of the system they are using |
| $T - 150 - 200g\sin\alpha = 200a$ or $F - T - 250 - 600g\sin\alpha = 600a$ | A1 | Unsimplified equation with at most one error |
| | A1 | Correct unsimplified equation |
| Obtain tension $206\text{(N)}$ or $210\text{(N)}$ | A1 | 3 sf or 2 sf only |
| **[4]** | | |

## Question 6a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments about $A$ | M1 | Dimensionally correct. Condone sine/cosine confusion |
| $5P = 40 \times \frac{7}{2}\cos\theta$ | A1 | Correct unsimplified equation |
| $P = 22.4$ | A1* | Obtain given answer from correct working. Need to see evidence of $\cos\theta = \frac{4}{5}$ |

**Total: [3]**

---

## Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| First equation | M1 | e.g. Resolve horizontally. Condone sine/cosine confusion |
| $H = P\sin\theta\ (=13.44)$ | A1 | Correct unsimplified equation |
| Second equation | M1 | e.g. Resolve vertically. Condone sine/cosine confusion |
| $V + P\cos\theta = 40\ (V = 22.08)$ | A1 | Correct unsimplified equation |
| $|R| = \sqrt{H^2 + V^2}$ | DM1 | Solve for $|R|$. Dependent on the 2 preceding Ms |
| $|R| = 26\ \text{(N)}$ | A1 | Or better $(25.84879\ldots)$. Accept $\frac{24\sqrt{29}}{5}$ |

**Total: [6]**

---

## Question 6b (alternative - resolve parallel/perpendicular):

| Answer/Working | Mark | Guidance |
|---|---|---|
| First equation | M1 | e.g. Resolve parallel. Condone sine/cosine confusion |
| $X = 40\sin\theta\ (=24)$ | A1 | Correct unsimplified equation |
| Second equation | M1 | e.g. Resolve perpendicular. Condone sine/cosine confusion |
| $Y + P = 40\cos\theta\ (Y = 9.6)$ | A1 | Correct unsimplified equation |
| $|R| = \sqrt{X^2 + Y^2}$ | DM1 | Solve for $|R|$. Dependent on the 2 preceding Ms |
| $|R| = 26\ \text{(N)}$ | A1 | Or better $(25.84879\ldots)$. Accept $\frac{24\sqrt{29}}{5}$ |

**Total: [6]**

Alternative moment equations:
- $M(C)$: $40 \times 1.5\cos\theta + H \times 5\sin\theta = V \times 5\cos\theta$
- $M(B)$: $2P + 7\cos\theta \times V = 7\sin\theta \times H + 3.5 \times 40\cos\theta$
- $M(G)$: $1.5P + 3.5\sin\theta \times H = 3.5\cos\theta \times V$

---

## Question 6b (alternative - cosine rule):

| Answer/Working | Mark | Guidance |
|---|---|---|
| 3 force diagram | M1 | 3 force diagram seen or implied |
| Forces and angle in correct positions | A1 | Forces and angle in correct positions |
| Use Cosine Rule | M1 | Correct formula used |
| $(|R|)^2 = 40^2 + 22.4^2 - 2 \times 40 \times 22.4\cos\theta$ | A1 | Correct unsimplified equation |
| Substitute for trig and solve for $|R|$ | DM1 | Dependent on the 2 preceding Ms |
| $|R| = 26\ \text{(N)}$ | A1 | Or better $(25.84879\ldots)$. Accept $\frac{24\sqrt{29}}{5}$ |

**Total: [6]**

---

## Question 7a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of CLM | M1 | Need all 4 terms. Dimensionally consistent. Condone sign errors. Condone $x$ in wrong direction |
| $6mu + 5mu = 5my - mx$ $(11u = 5y - x)$ | A1 | Correct unsimplified equation |
| Use of impact law | M1 | Used correctly. Dimensionally correct. Condone sign errors |
| $x + y = 5eu$ | A1 | Correct unsimplified equation. Signs consistent with CLM equation |
| Solve for $x$: $6x = 25eu - 11u$ or solve for $e$: $e = \frac{6y-11u}{5u}$ | DM1 | Dependent on first 2 M marks. As far as $kx = \ldots$ Dependent on previous 2 M marks |
| Use $x > 0\ (\Rightarrow y > \frac{11}{5}u)$: $25e > 11$ | DM1 | Use correct inequality for their $x$ |
| $\frac{11}{25} < e\ (\text{,}\ 1)$ | A1 | Or equivalent. Condone if 1 not mentioned. Allow with $<1$. A0 if incorrect upper limit. |

**Total: [7]**

---

## Question 7b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{2}{3}u$ and $y = \frac{7}{3}u$ | B1 | Seen or implied |
| Total KE lost $= \left(\frac{1}{2}m \times 36u^2 + \frac{1}{2}5m \times u^2\right) - \left(\frac{1}{2}m \times x^2 + \frac{1}{2}5m \times y^2\right)$ | M1 | Complete expression. Dimensionally correct. Correct masses connected to correct speeds. Condone subtraction in wrong order. Allow in $x$ and $y$ |
| $= \left(\frac{1}{2}m \times 36u^2 + \frac{1}{2}5m \times u^2\right) - \left(\frac{1}{2}m \times \frac{4}{9}u^2 + \frac{1}{2}5m \times \frac{49}{9}u^2\right)$ | A1ft | Correct unsimplified expression in $m$ and $u$. Follow their $x$, $y$ with $e$ substituted |
| $= \frac{20}{3}mu^2$ | A1 | Or single term equivalent. Accept $6.7mu^2$ or better |

**Total: [4]**

---

## Question 7c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Velocity of $Q$ after collision with wall $= \pm fy\ \left(= \pm f \times \frac{7}{3}u\right)$ | B1ft | Follow their $y$ (in terms of $u$) |
| Second collision if $fy > x$: $\frac{7}{3}fu > \frac{2}{3}u$ | DM1 | Correct inequality for their $x$, $y$. Dependent on B1 and $P$ moving away from wall |
| $\frac{2}{7} < f\ \text{,}\ 1$ | A1 | Correct only. Need both limits |

**Total: [3]**

---

## Question 8a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use symmetry to find time taken: $-7 = 7 - gt$ | M1 | Or equivalent complete method using suvat to find time e.g. find time for vertical distance $= 0$ |
| $t = \frac{14}{g}\ (=1.428\ldots)$ | A1 | Correct value seen or implied |
| Horizontal distance $= 4t$ | DM1 | Complete method using suvat to find distance. Dependent on preceding M1 |
| $= 5.71\ \text{(m)}$ or $5.7\ \text{(m)}$ | A1 | 3 sf or 2 sf only. $\frac{40}{7}$ scores A0. $\frac{56}{g}$ scores A0 (incorrect units) |

**Total: [4]**

---

## Question 8a (alternative):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Find speed and angle of projection | M1 | Correct use of Pythagoras and trig |
| Speed $= \sqrt{16+49} = \sqrt{65}\ (\text{ms}^{-1})$ | A1 | Both values seen or implied |
| Direction $= \tan^{-1}\frac{7}{4}\ (= 60.3°)$ | | |
| Use of $R = \frac{u^2\sin 2\alpha}{g}$ | DM1 | Or equivalent. Dependent on preceding M1 |
| $= 5.71\ \text{(m)}$ or $5.7\ \text{(m)}$ | A1 | 3 sf or 2 sf only |

**Total: [4]**

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## Question 8b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $|\mathbf{v}| = 5 \Rightarrow \mathbf{v} = 4\mathbf{i} + 3\mathbf{j}$ or $\mathbf{v} = 4\mathbf{i} - 3\mathbf{j}$ | B1 | Correct vertical component seen or implied |
| $-3 = 3 - gT$ | M1 | Complete method to find $T$. e.g. $T = \frac{14}{g} - 2 \times \frac{4}{g}$ |
| $T = 0.612$ or $T = 0.61$ | A1 | 3 sf or 2 sf only. $\frac{30}{49}$ scores A0. $\frac{6}{g}$ scores A0 (incorrect units) |

**Total: [3]**

---

## Question 8c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}4\\7\end{pmatrix} \cdot \begin{pmatrix}4\\p\end{pmatrix} = 0$ | M1 | Or equivalent method to find perpendicular velocity |
| $\Rightarrow p = -\frac{16}{7}$, $\mathbf{v} = 4\mathbf{i} - \frac{16}{7}\mathbf{j}$ | A1 | Correct vertical component. Allow $-2.28\ldots$ |
| $\left(-\frac{16}{7}\right)^2 = 7^2 - 2gh$ | DM1 | Complete method using suvat or energy to form equation in $h$ only. Dependent on preceding M1 |
| $h = 2.23$ or $h = 2.2$ | A1 | 3 sf or 2 sf only cso (negative vertical component seen at some point) |

**Total: [4]**

---

## Question 8c (alternative):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}4\\7\end{pmatrix} \cdot \begin{pmatrix}4\\7-gt\end{pmatrix} = 0$ | M1 | Or equivalent method to find time when velocity perpendicular |
| $t = \frac{65}{7g}\ (= 0.947\ldots)$ | A1 | Correct time |
| $h = 7t - \frac{1}{2}gt^2$ | DM1 | Complete method using suvat to form equation in $h$ only |
| $h = 2.23$ or $h = 2.2$ | A1 | 3 sf or 2 sf only cso |

**Total: [4]**