| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Direct collision with direction reversal |
| Difficulty | Standard +0.3 This is a standard M2 collision problem requiring conservation of momentum and Newton's experimental law (restitution). Parts (a) and (b) are routine algebraic manipulations with clear sign conventions. Part (c) requires combining the results to find bounds on k, which is slightly more sophisticated but still follows a standard template. The multi-part structure and 9 marks indicate moderate length, but no novel insight is required—this is textbook collision mechanics. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Momentum: \(mu - kmu = -m\frac{u}{3} + kmv\) | M1 M1 A1 | |
| \(\frac{6u}{5} - ku = kv\) | ||
| \(v = u\left(\frac{6}{5k} - 1\right)\) | A1 | |
| (b) Elasticity: \((v + \frac{u}{3})/(u - u) = -e\) | M1 A1 | |
| \(v = 2eu - \frac{u}{3} = u\left(2e - \frac{1}{3}\right)\) | M1 A1 | |
| (c) \(v > 0\), so \(\frac{6}{5k} > 1\) | M1 M1 A1 | |
| \(k < \frac{6}{5}\) | ||
| Also \(2e > \frac{1}{3}\), so \(e > \frac{1}{10}\) | M1 M1 A1 | |
| Hence \(\frac{1}{10} < e \leq 1\), so \(0 < v < \frac{9u}{5}\) | M1 A1 | |
| \(0 < \frac{6}{5k} - 1 < \frac{2}{5}\) | M1 | |
| \(\frac{6}{5k} \leq \frac{7}{5}\) | ||
| \(14k \geq 6\) | ||
| \(k \geq \frac{3}{7}\) | A1 A1 A1 | |
| \(p = \frac{4}{7}\), \(q = \frac{6}{5}\) | M1 A1 A1 | 17 marks |
(a) Momentum: $mu - kmu = -m\frac{u}{3} + kmv$ | M1 M1 A1 |
$\frac{6u}{5} - ku = kv$ | |
$v = u\left(\frac{6}{5k} - 1\right)$ | A1 |
(b) Elasticity: $(v + \frac{u}{3})/(u - u) = -e$ | M1 A1 |
$v = 2eu - \frac{u}{3} = u\left(2e - \frac{1}{3}\right)$ | M1 A1 |
(c) $v > 0$, so $\frac{6}{5k} > 1$ | M1 M1 A1 |
$k < \frac{6}{5}$ | |
Also $2e > \frac{1}{3}$, so $e > \frac{1}{10}$ | M1 M1 A1 |
Hence $\frac{1}{10} < e \leq 1$, so $0 < v < \frac{9u}{5}$ | M1 A1 |
$0 < \frac{6}{5k} - 1 < \frac{2}{5}$ | M1 |
$\frac{6}{5k} \leq \frac{7}{5}$ | |
$14k \geq 6$ | |
$k \geq \frac{3}{7}$ | A1 A1 A1 |
$p = \frac{4}{7}$, $q = \frac{6}{5}$ | M1 A1 A1 | 17 marks8. Two ships $A$ and $B$, of masses $m$ and km respectively, are moving towards each other in heavy fog along the same straight line, both with speed $u$. The ships collide and immediately after the collision they drift away from each other, both their directions of motion having been reversed. The speed of $A$ after the impact is $\frac { 1 } { 5 } u$ and the speed of $B$ after the impact is $v$.
\begin{enumerate}[label=(\alph*)]
\item Show that $v = u \left( \frac { 6 } { 5 k } - 1 \right)$.
The coefficient of restitution between $A$ and $B$ is $e$.
\item Show that $v = u \left( 2 e - \frac { 1 } { 5 } \right)$.
\item Use your answers to parts (a) and (b) to find the rational numbers $p$ and $q$ such that $p \leq k < q$.\\
(9 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q8 [17]}}