| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2005 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Standard +0.3 This is a standard M2 momentum-collision problem requiring conservation of momentum and Newton's restitution law applied twice. Part (a) is routine calculation, part (b) requires comparing velocities after the second collision to verify no further collisions occur—straightforward but involves multiple steps and careful bookkeeping of directions. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| CLM: \(6mu - 4mu = 3mv + 4mu\) | M1 A1 | |
| \(\Rightarrow v = -\frac{2}{3}u\) | A1 | |
| NLI: \(2u - v = e \cdot 4u\) | M1 A1 | |
| \(\Rightarrow 4eu = \frac{8}{3}u \Rightarrow e = \frac{2}{3}\) | M1 A1 | |
| (7) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(5my + 2mx = 4mu\) | M1 A1 | |
| \(y - x = \frac{3}{5} \cdot 2u = \frac{6}{5}u\) | A1 | |
| Solve: \(x = -\frac{2}{7}u\) | M1 A1 | |
| \(\frac{2}{7}u < \frac{2}{3}u\) so \(B\) does not overtake \(A\) | M1 | |
| So no more collisions | A1 cso | |
| (7) |
## Question 5:
### Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| CLM: $6mu - 4mu = 3mv + 4mu$ | M1 A1 | |
| $\Rightarrow v = -\frac{2}{3}u$ | A1 | |
| NLI: $2u - v = e \cdot 4u$ | M1 A1 | |
| $\Rightarrow 4eu = \frac{8}{3}u \Rightarrow e = \frac{2}{3}$ | M1 A1 | |
| | **(7)** | |
### Part (b):
| Working | Marks | Notes |
|---------|-------|-------|
| $5my + 2mx = 4mu$ | M1 A1 | |
| $y - x = \frac{3}{5} \cdot 2u = \frac{6}{5}u$ | A1 | |
| Solve: $x = -\frac{2}{7}u$ | M1 A1 | |
| $\frac{2}{7}u < \frac{2}{3}u$ so $B$ does not overtake $A$ | M1 | |
| So no more collisions | A1 cso | |
| | **(7)** | |
---5. Two small spheres $A$ and $B$ have mass $3 m$ and $2 m$ respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed $2 u$, when they collide directly. As a result of the collision, the direction of motion of $B$ is reversed and its speed is unchanged.
\begin{enumerate}[label=(\alph*)]
\item Find the coefficient of restitution between the spheres.
Subsequently, $B$ collides directly with another small sphere $C$ of mass $5 m$ which is at rest. The coefficient of restitution between $B$ and $C$ is $\frac { 3 } { 5 }$.
\item Show that, after $B$ collides with $C$, there will be no further collisions between the spheres.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2005 Q5 [14]}}