| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2004 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring conservation of momentum, Newton's restitution law, and analysis of conditions for a second collision. Part (a) is standard M2 fare, but parts (b) and (c) require careful tracking of velocities through multiple collisions and inequality reasoning to determine when spheres meet again, which elevates it above routine exercises. |
| Spec | 6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| LM: \(10mu = 2mx + 3my\) | M1 A1 | |
| NEL: \(y - x = 5eu\) | B1 | |
| Solving to \(y = 2(1+e)u\) | M1 A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 2u - 3eu\); finding \(x\), with or without \(e = 0.4\) | M1 | |
| \(x = 0.8u\) | A1 | |
| \(x > 0 \Rightarrow P\) moves towards wall and \(Q\) rebounds from wall \(\Rightarrow\) second collision | A1ft | ft any positive \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = -0.4u\) | B1 | |
| Speed of \(Q\) on rebound is \(3.6fu\) | ||
| For second collision: \(3.6fu > 0.4u\) | M1 | |
| \(f > \dfrac{1}{9}\) | A1 | ignore \(f \mid 1\) |
## Question 5:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| LM: $10mu = 2mx + 3my$ | M1 A1 | |
| NEL: $y - x = 5eu$ | B1 | |
| Solving to $y = 2(1+e)u$ | M1 A1 | cso |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 2u - 3eu$; finding $x$, with or without $e = 0.4$ | M1 | |
| $x = 0.8u$ | A1 | |
| $x > 0 \Rightarrow P$ moves towards wall and $Q$ rebounds from wall $\Rightarrow$ second collision | A1ft | ft any positive $x$ |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = -0.4u$ | B1 | |
| Speed of $Q$ on rebound is $3.6fu$ | | |
| For second collision: $3.6fu > 0.4u$ | M1 | |
| $f > \dfrac{1}{9}$ | A1 | ignore $f \mid 1$ |
---5. Two small smooth spheres, $P$ and $Q$, of equal radius, have masses $2 m$ and $3 m$ respectively. The sphere $P$ is moving with speed $5 u$ on a smooth horizontal table when it collides directly with $Q$, which is at rest on the table. The coefficient of restitution between $P$ and $Q$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $Q$ immediately after the collision is $2 ( 1 + e ) u$.
After the collision, $Q$ hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of $Q$. The coefficient of restitution between $Q$ and the wall is $f , 0 < f \leqslant 1$.
\item Show that, when $e = 0.4$, there is a second collision between $P$ and $Q$.
Given that $e = 0.8$ and there is a second collision between $P$ and $Q$,
\item find the range of possible values of $f$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2004 Q5 [11]}}