| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.3 This is a standard M2 collision problem requiring conservation of momentum and Newton's restitution law, followed by straightforward inequality manipulation and direction analysis. While it has multiple parts and requires careful sign conventions, the techniques are routine for M2 students with no novel problem-solving insight needed. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks |
|---|---|
| Momentum: \(4mu - 5mu = mv_P + mv_{Q}\) | M1 A1 |
| \(v_P + v_Q = -u\) | |
| Elasticity: \((v_Q - v_P)(-5u - 4u) = -e\) | M1 A1 |
| \(v_Q - v_P = 9eu\) | |
| Add: \(2v_Q = 9eu - u\) | M1 A1 |
| \(v_Q = \frac{1}{2}(9e - 1)u\) |
| Answer | Marks |
|---|---|
| \(v_Q > 0\), so \(9e > 1\) | M1 A1 |
| \(e > \frac{1}{9}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_P = -\frac{1}{2}(9e + 1)u\) | M1 A1 | |
| After hitting wall, speed of \(Q < \frac{1}{2}(9e - 1)u\) | M1 A1 | |
| which is clearly less than \( | v_P | \), so there is no further collision |
### (a)
Momentum: $4mu - 5mu = mv_P + mv_{Q}$ | M1 A1 |
$v_P + v_Q = -u$ | |
Elasticity: $(v_Q - v_P)(-5u - 4u) = -e$ | M1 A1 |
$v_Q - v_P = 9eu$ | |
Add: $2v_Q = 9eu - u$ | M1 A1 |
$v_Q = \frac{1}{2}(9e - 1)u$ | |
### (b)
$v_Q > 0$, so $9e > 1$ | M1 A1 |
$e > \frac{1}{9}$ | |
### (c)
$v_P = -\frac{1}{2}(9e + 1)u$ | M1 A1 |
After hitting wall, speed of $Q < \frac{1}{2}(9e - 1)u$ | M1 A1 |
which is clearly less than $|v_P|$, so there is no further collision | A1 | **Total: 11 marks**Two railway trucks, $P$ and $Q$, of equal mass, are moving towards each other with speeds $4u$ and $5u$ respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between $P$ and $Q$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $u$ and $e$, the speed of $Q$ after the collision. [6 marks]
\item Show that $e > \frac{1}{9}$. [2 marks]
\end{enumerate}
$Q$ now hits a fixed buffer and rebounds along the track. $P$ continues to move with the speed that it had immediately after it collided with $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Prove that it is impossible for a further collision between $P$ and $Q$ to occur. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q6 [11]}}