# Question 5:

## Part 5a:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $P = Fv\ \left(F = \dfrac{500}{6}\right)$ | M1 | |
| Equation of motion | M1 | Dimensionally correct. Required terms and no extras |
| $F - 60 = 80a$ | A1 | Correct unsimplified equation in $F$ |
| $a = \dfrac{7}{24}\ (\text{ms}^{-2})$ | A1 | 0.29 or better (0.291666666..) |

**Total: [4]**

## Part 5b:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Gain in $\text{KE} = \frac{1}{2} \times 80 \times 8^2\ \text{(J)}\ (= 2560\ \text{J})$ | B1 | Any one correct (seen or implied) |
| Gain in $\text{GPE} = 80 \times 9.8 \times 300\ \text{(J)}\ (= 235200\ \text{J})$ | B1 | A second term correct (seen or implied) |
| Work done against resistance $= 20000 \times 60$ | | $(\text{KE gain} + \text{GPE gain} = 237760\ \text{J})$ |
| **Use of *suvat* and $F = ma$ is M0A0A0** | | |
| Expression for combined work and energy | M1 | All terms required and no double counting. Mass replaced with 80. Condone sign errors. Dimensionally correct. Condone error in zeros in 20000 |
| Total work done $= 40\times 64 + 80\times 9.8\times 300 + 20000\times 60$ | A1 | Correct unsimplified expression for work done |
| $1440\ \text{(kJ)}$ or $1400\ \text{(kJ)}$ | A1 | Accept answers in joules. 3 sf or 2 sf (1437760) |

**Total: [5]**

## Part 5c:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of motion | M1 | Dimensionally correct. Required terms and no extras |
| $F - 60 - 80g \times \sin\alpha = 0$ | A1 | Unsimplified equation in $P$ or $F$ with at most one error |
| $\dfrac{P}{7} - 60 - 80g \times \dfrac{1}{20} = 0$ | A1 | Correct unsimplified equation in $P$ |
| $P = 694$ or $P = 690$ | A1 | 3 sf or 2 sf only |

**Total: [4] | Overall: (13)**

## Question 6a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments about $A$: **M0 if there is no resolving** | M1 | Need all terms and no extras. Dimensionally consistent. Condone sign errors and sine/cosine confusion. |
| $4a\cos 30°\times W + 8a\cos 30°\times\dfrac{W}{4} = 5a\cos 30°\times T$ | A1 | Correct unsimplified equation |
| $6W = 5T \Rightarrow T = \dfrac{6}{5}W$ * | A1* | Obtain given answer from correct working, e.g. show cancelling of the common factors or some simplification of the moments equation |

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## Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| First equation dimensionally correct. Condone sine/cosine confusion and sign errors | M1 | e.g. Resolve horizontally |
| $H = T\cos 30°$ | A1 | $H = \dfrac{3\sqrt{3}}{5}W$ |
| Second equation dimensionally correct. Condone sine/cosine confusion and sign errors | M1 | e.g. Resolve vertically |
| $V + T\cos 60° = W + \dfrac{W}{4}$ | A1 | $V = \dfrac{13}{20}W$ |
| $\|R\| = \sqrt{V^2 + H^2}$ or $\|R\|^2 = V^2 + H^2$ | DM1 | Correct use of Pythagoras. Dependent on two preceding M marks. |
| $\|R\| = \dfrac{W}{20}\sqrt{3\times144 + 169} = \dfrac{\sqrt{601}}{20}W$ | A1 | $1.2W$ or better $(1.22576\ldots)$ |

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## Question 7a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation for CLM | M1 | Dimensionally correct. All terms required. Condone sign errors. |
| $8mu - 6mu = 2my - 4mx$ giving $(u = y - 2x)$ | A1 | Correct unsimplified equation |
| Equation for kinetic energy $\left(\frac{1}{2}\text{ or }2\text{ must be used}\right)$ | M1 | Dimensionally correct. Correct masses paired with correct velocities. All terms required. No sign errors. Condone 2 on the wrong side. |
| $2mx^2 + my^2 = \frac{1}{2}\left(2m\times 4u^2 + m\times 9u^2\right)$ giving $(17u^2 = 4x^2 + 2y^2)$ | A1 | Correct unsimplified equation |
| Solve for $y$: $17u^2 = 2y^2 + (y-u)^2 \Rightarrow 3y^2 - 2yu - 16u^2 = 0$ | DM1 | Some working must be shown to obtain the quadratic in $y$ (and $u$). Dependent on preceding M marks. $\left((3y-8u)(y+2u)=0\right)$ |
| $\Rightarrow y = \dfrac{8}{3}u$ * | A1* | Obtain given answer from correct working |

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## Question 7b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of Impact Law: $x + y = e\times 5u$ | M1 | Condone sign errors but must be used the right way round. |
| $e = \dfrac{\frac{1}{2}\left(\frac{8}{3}u - u\right) + \frac{8}{3}u}{5u}$ | A1 | Correct unsimplified equation. $\left(x = \dfrac{5u}{6}\right)$ |
| $= \dfrac{7}{10}$ | A1 | Correct only |

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## Question 7c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Velocity of $Q$ after impact $= f\times\dfrac{8}{3}u$ | B1 | Allow $\pm$ |
| No collision if $f\times\dfrac{8}{3}u \leq \dfrac{5}{6}u$, i.e. speed of $P \geq$ speed of $Q$ | M1 | Correct inequality with their values. Accept strict inequality. Dimensionally correct. |
| $\Rightarrow 0 < f \leq \dfrac{5}{16}$ | A1 | Both ends required. $(0 < f \leq 0.3125)$ |

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## Question 7d:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $I = \pm 2m\left(y - \left(-\dfrac{1}{4}y\right)\right)$ | M1 | Subtraction seen or implied with their $\frac{1}{4}y$. Requires correct mass. Requires correct impact law. |
| $\|I\| = \dfrac{20}{3}mu$ | A1 | Or equivalent. Must be positive. $6.7mu$ or better. Condone $-\dfrac{20}{3}mu \rightarrow \dfrac{20}{3}mu$ with no explanation. |