## 7a.
CLM: $2mu = 2mv + 3mw$ | M1 A1 |
Impact law: $eu = w - v$ | M1 A1 |
Eliminate w: $5v = u(2 - 3e)$ oe | M1 |
$v = \frac{u}{3}$ | M1 |
$e = \frac{1}{9}, 0.11$ or better | A1 |

**OR:** $v = \frac{u}{3}$ is used in CLM and NIL equations (See Alternative below) | |

**ALTERNATIVE:**
$v = \frac{u}{3}$ | M1 (KE) |
$2mu = 2mv + 3mw$ ($2mu = 2m\frac{u}{3} + 3mw$) | M1 A1 (CLM) |
$\Rightarrow w = \frac{4u}{9}$ | M1 (eliminate w) |
$e = \frac{w - v}{u} = \left(e = \frac{\frac{4u}{9} - \frac{u}{3}}{u}\right)$ | M1 A1 (Impact Law) |
$e = \frac{1}{9}$ | A1 |

## 7b(i).
Substitute $e\left(= \frac{1}{9}\right): v = \frac{u}{3}, w = \frac{4}{9}u$ | M1 |
CLM & Impact equations: $3m \times w = 3mV + 4mW$ ($\frac{4}{3}u = 3V + 4W$) | M1 |
$w f = W - V$ ($\frac{16u}{9}f = 4W - 4V$) | A1 |
Follow their w | |
Solve for $V$: $V = \frac{4u}{21} - \frac{16uf}{63}$ | M1 |
N.B. answer must be positive | A1 |

**since $f \geq 0$, $V \leq \frac{4u}{21} < \frac{u}{3}$ and hence A collides with B** | M1 A1 |

## 7b(ii).
$(ii)$
Fourth M1 (only available if they have a correct expression for $V$) for a complete strategy to show that there is a 2$^{nd}$ collision between A and B, using $f \geq 0$ (Allow M1 A0 if they use $f = 0$, see above)
Third A1 for a fully correct justification. N.B. GIVEN ANSWER

**(ii) Alternative:**
Fourth M1 (only available if they have a correct expression for $V$) for a complete strategy to show that there is a 2$^{nd}$ collision between A and B:

2$^{nd}$ collision if $V < \frac{u}{3}$

i.e. if $\frac{4u}{21} - \frac{16uf}{63} < \frac{u}{3}$

i.e. if $\frac{-9}{16} < f$.

Since $f \geq 0$ (given), there will be a 2$^{nd}$ collision.
Third A1 for a fully correct justification. N.B. GIVEN ANSWER.

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# Notes and General Information:

- Follow through (ft) marks are indicated where appropriate
- DM indicates a dependent mark that requires a previous mark to have been earned
- A marks can be 'follow through' (A ft) when they depend on previous working
- CAO indicates "correct answer only"
- CSO indicates "correct solution only" with no errors
- Allow indicates acceptable variations