| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Particle brought to rest by collision |
| Difficulty | Standard +0.8 This is a multi-part collision problem requiring conservation of momentum, coefficient of restitution, and impulse-momentum theorem across two collisions. Part (a) is standard M2 fare, but parts (b) and (c) require algebraic manipulation to show specific results, with part (c) demanding insight that terms cancel to make the final answer independent of k—a non-trivial result requiring careful symbolic work across multiple equations. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks |
|---|---|
| (a) Momentum: \(3mu = 3mv_A + 4mv_B\), \(3v_A + 4v_B = 3u\) | M1 A1 |
| Elasticity: \((v_B - v_A)/(-u) = -e\), \(3v_B - 3v_A = 3eu\) | M1 A1 |
| Add: \(3u(1+e) = 7v_B\), \(v_B = \frac{3}{7}u(1+e)\) | M1 A1 |
| (b) If \(v_A = 0\), \(v_B = eu\) and \(4v_B = 3u\), so \(e = 0.75\) | M1 A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(ku + 3u = 3v_A' + 4(v_A' - 0.75(0.75 - \frac{1}{3}k)u) = 7v_A' - 2.25u + ku\) | M1 A1 M1 A1 M1 A1 | |
| \(v_A' = 0.75u\), which is independent of \(k\) | A1 | Total: 16 marks |
(a) Momentum: $3mu = 3mv_A + 4mv_B$, $3v_A + 4v_B = 3u$ | M1 A1 |
Elasticity: $(v_B - v_A)/(-u) = -e$, $3v_B - 3v_A = 3eu$ | M1 A1 |
Add: $3u(1+e) = 7v_B$, $v_B = \frac{3}{7}u(1+e)$ | M1 A1 |
(b) If $v_A = 0$, $v_B = eu$ and $4v_B = 3u$, so $e = 0.75$ | M1 A1 A1 |
(c) Now 4 has speed $\frac{1}{3}ku$
$(v_B' - v_A')/(0.75u - \frac{1}{3}ku) = -0.75$
and $kmu + 3mu = 3mv_A' + 4mv_B'$
$ku + 3u = 3v_A' + 4(v_A' - 0.75(0.75 - \frac{1}{3}k)u) = 7v_A' - 2.25u + ku$ | M1 A1 M1 A1 M1 A1 |
$v_A' = 0.75u$, which is independent of $k$ | A1 | Total: 16 marksTwo smooth spheres, $A$ and $B$, of equal radius but of masses $3m$ and $4m$ respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is $e$.
$A$ is projected with speed $u$ to hit $B$, which is initially at rest.
\begin{enumerate}[label=(\alph*)]
\item Show that $B$ begins to move with speed $\frac{3}{7}u(1 + e)$. [6 marks]
\item Given that $A$ is brought to rest by the collision, show that $e = 0.75$. [3 marks]
\end{enumerate}
Having been brought to rest, $A$ is now set in motion again by being given an impulse of magnitude $kmu$ Ns, where $k > 2.25$. $A$ then collides again with $B$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that the speed of $A$ after this second impact is independent of $k$. [7 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q7 [16]}}