| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Particle brought to rest by collision |
| Difficulty | Standard +0.3 This is a standard M2 collision problem requiring application of conservation of momentum and Newton's restitution law. Part (a) involves straightforward algebraic manipulation of two equations to show a given result. Part (b) requires simple inequality reasoning using the constraint 0 < e ≤ 1 and 0 < λ < 1/2. While it involves multiple steps, the techniques are routine for M2 students and the 'show that' format provides the target answer. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(mu + km\lambda u = kmv\) | M1 A1 | Conservation of momentum |
| \(\frac{u}{k}(1 + k\lambda) = v\) | ||
| \(v = e(u - \lambda u) = eu(1-\lambda)\) | M1 A1 | Newton's law of restitution |
| \(\Rightarrow \frac{u}{k}(1+k\lambda) = eu(1-\lambda)\) | M1 | |
| \(\Rightarrow \dfrac{(1+k\lambda)}{k(1-\lambda)} = e\) | A1 | (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{1+k\lambda}{k(1-\lambda)} \leq 1 \Rightarrow 1 + k\lambda \leq k(1-\lambda)\) | M1 | |
| \(\Rightarrow \dfrac{1}{1-2\lambda} \leq k\) | A1 | |
| since \(0 < \lambda < \frac{1}{2},\ k > 1\) | A1 | (3) (9) |
# Question 5:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $mu + km\lambda u = kmv$ | M1 A1 | Conservation of momentum |
| $\frac{u}{k}(1 + k\lambda) = v$ | | |
| $v = e(u - \lambda u) = eu(1-\lambda)$ | M1 A1 | Newton's law of restitution |
| $\Rightarrow \frac{u}{k}(1+k\lambda) = eu(1-\lambda)$ | M1 | |
| $\Rightarrow \dfrac{(1+k\lambda)}{k(1-\lambda)} = e$ | A1 | (6) |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{1+k\lambda}{k(1-\lambda)} \leq 1 \Rightarrow 1 + k\lambda \leq k(1-\lambda)$ | M1 | |
| $\Rightarrow \dfrac{1}{1-2\lambda} \leq k$ | A1 | |
| since $0 < \lambda < \frac{1}{2},\ k > 1$ | A1 | (3) **(9)** |
---5. A smooth sphere $S$ of mass $m$ is moving with speed $u$ on a smooth horizontal plane. The sphere $S$ collides with another smooth sphere $T$, of equal radius to $S$ but of mass $k m$, moving in the same straight line and in the same direction with speed $\lambda u , 0 < \lambda < \frac { 1 } { 2 }$. The coefficient of restitution between $S$ and $T$ is $e$.
Given that $S$ is brought to rest by the impact,
\begin{enumerate}[label=(\alph*)]
\item show that $e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }$.
\item Deduce that $k > 1$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q5 [9]}}