| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Successive collisions, three particles in line |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring systematic application of conservation of momentum and Newton's restitution law across three separate collisions, with careful tracking of velocities and directions. Part (a) is standard M2 fare (showing a given result), but parts (b) and (c) require students to analyze whether subsequent collisions occur by comparing velocities after each impact—a more sophisticated problem-solving task than typical textbook exercises. The multiple collisions and need to track which sphere moves where elevates this above average difficulty. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| CLM: \(mv + 5mw = mu\) | B1 | Signs consistent with diagram |
| NLI: \(w - v = eu\) | B1 | Impact equation |
| Solve \(v\): \(v = \frac{1}{6}(1-5e)u\), speed \(= \frac{1}{6}(5e-1)u\) | M1\* A1 | Answer given; attempt to eliminate \(w\); correct expression for \(v\); must verify positive with reference to \(e > \frac{1}{5}\) |
| Solve \(w\): \(w = \frac{1}{6}(1+e)u\) | M1\* A1 | Attempt to eliminate \(v\); correct expression for \(w\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| After \(B\) hits \(C\), velocity of \(B\) = \(\frac{1}{6}(1 - 5 \cdot \frac{4}{5})u = -\frac{1}{2}u\) | M1 A1 | Substitute for \(e\) in speed/velocity of \(B\) to obtain \(v\) in terms of \(u\) |
| velocity \(< 0 \Rightarrow\) change of direction \(\Rightarrow B\) hits \(A\) | A1 CSO | Justify direction and correct conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Velocity of \(C\) after \(= \frac{3}{10}u\) | B1 | Speed of \(C\) = value of \(w = (\pm)\frac{3u}{10}\); must be referred to in (c) |
| When \(B\) hits \(A\), \(u'' = \frac{1}{2}u\), so velocity of \(B\) after \(= -\frac{1}{2}(-\frac{1}{2}u) = \frac{1}{4}u\) | B1 | Speed of \(B\) after second collision \((\pm)\frac{1}{4}u\) or \((\pm)\frac{5}{6}w\) |
| Travelling in same direction but \(\frac{1}{4} < \frac{3}{10} \Rightarrow\) no second collision | M1 A1 CSO | Comparing speed of \(B\) after 2nd collision with speed of \(C\) after first collision; correct conclusion |
# Question 7:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| CLM: $mv + 5mw = mu$ | B1 | Signs consistent with diagram |
| NLI: $w - v = eu$ | B1 | Impact equation |
| Solve $v$: $v = \frac{1}{6}(1-5e)u$, speed $= \frac{1}{6}(5e-1)u$ | M1\* A1 | Answer given; attempt to eliminate $w$; correct expression for $v$; must verify positive with reference to $e > \frac{1}{5}$ |
| Solve $w$: $w = \frac{1}{6}(1+e)u$ | M1\* A1 | Attempt to eliminate $v$; correct expression for $w$ |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| After $B$ hits $C$, velocity of $B$ = $\frac{1}{6}(1 - 5 \cdot \frac{4}{5})u = -\frac{1}{2}u$ | M1 A1 | Substitute for $e$ in speed/velocity of $B$ to obtain $v$ in terms of $u$ |
| velocity $< 0 \Rightarrow$ change of direction $\Rightarrow B$ hits $A$ | A1 CSO | Justify direction and correct conclusion |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Velocity of $C$ after $= \frac{3}{10}u$ | B1 | Speed of $C$ = value of $w = (\pm)\frac{3u}{10}$; must be referred to in (c) |
| When $B$ hits $A$, $u'' = \frac{1}{2}u$, so velocity of $B$ after $= -\frac{1}{2}(-\frac{1}{2}u) = \frac{1}{4}u$ | B1 | Speed of $B$ after second collision $(\pm)\frac{1}{4}u$ or $(\pm)\frac{5}{6}w$ |
| Travelling in same direction but $\frac{1}{4} < \frac{3}{10} \Rightarrow$ no second collision | M1 A1 CSO | Comparing speed of $B$ after 2nd collision with speed of $C$ after first collision; correct conclusion |
---\begin{enumerate}
\item Two small spheres $P$ and $Q$ of equal radius have masses $m$ and $5 m$ respectively. They lie on a smooth horizontal table. Sphere $P$ is moving with speed $u$ when it collides directly with sphere $Q$ which is at rest. The coefficient of restitution between the spheres is $e$, where $e > \frac { 1 } { 5 }$.\\
(a) (i) Show that the speed of $P$ immediately after the collision is $\frac { u } { 6 } ( 5 e - 1 )$.\\
(ii) Find an expression for the speed of $Q$ immediately after the collision, giving your answer in the form $\lambda u$, where $\lambda$ is in terms of $e$.\\
(6)
\end{enumerate}
Three small spheres $A , B$ and $C$ of equal radius lie at rest in a straight line on a smooth horizontal table, with $B$ between $A$ and $C$. The spheres $A$ and $C$ each have mass $5 m$, and the mass of $B$ is $m$. Sphere $B$ is projected towards $C$ with speed $u$. The coefficient of restitution between each pair of spheres is $\frac { 4 } { 5 }$.\\
(b) Show that, after $B$ and $C$ have collided, there is a collision between $B$ and $A$.\\
(c) Determine whether, after $B$ and $A$ have collided, there is a further collision between $B$ and $C$.\\
\hfill \mbox{\textit{Edexcel M2 2007 Q7 [13]}}