## Question 4:

**Part (a):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Conservation of momentum | M1 | Correct no. of terms, allow consistent cancelled $m$'s: $\left(u + e^2u = v_P + ev_Q\right)$ |
| $mu + e^2mu = mv_P + emv_Q$ | A1 | Correct unsimplified equation |
| Newton's Impact Law | M1 | Correct no. of terms, with $e$ on correct side |
| $e(u - eu) = -v_P + v_Q$ | A1 | Correct unsimplified equation |
| Solve these equations for $v_Q$ | M1 | |
| $v_Q = u^*$ | A1* | |

**Part (b):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $v_P = u(e^2 - e + 1) \left(= \frac{(e^3+1)u}{e+1}\right)$ | M1 | |
| $= u\left(\left(e - \frac{1}{2}\right)^2 + \frac{3}{4}\right)$ | A1 | |
| $> 0$ so $P$ continues to move in the same direction* | A1* | |

**Part (c):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use impulse-momentum principle | M1 | |
| $I = em(u - eu)$ or $m(-u(e^2 - e+1) - (-u))$ $(= (e - e^2)mu)$ | A1 | |
| $(e - e^2) = \frac{2}{9}$ and solve | M1 | |
| $e = \frac{1}{3}$ or $\frac{2}{3}$ | A1 | |

## Question 4b:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Solve for $v_p$ | M1 | |
| Completing the square or any other appropriate method | M1 | |
| Correct conclusion correctly reached | A1* | cao |

## Question 4c:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Correct no. of terms, dimensionally correct expression in terms of $e$ and $u$ | M1 | Must be subtracting. Needs to be in terms of $e$ and $u$ |
| Correct unsimplified expression | A1 | Allow negative answer at this stage |
| Solving an appropriate quadratic equation | M1 | |
| Two correct answers | A1 | |