| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Range of coefficient of restitution |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring conservation of momentum, coefficient of restitution, and analysis of conditions for subsequent collisions. Part (a) is standard M2 fare (showing a given result), part (b) requires energy calculations, but part (c) demands sophisticated reasoning about velocity conditions across two collisions to determine when a second P-Q collision occurs—this requires algebraic manipulation and inequality analysis beyond typical textbook exercises. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Solving to \(y = \frac{8}{5}u\) | M1 A1 B1 M1 A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{9}{20}mu^2\) | B1 M1 A(2,1,0) A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(e > \frac{1}{4}\) ignore \(e \leq 1\) | M1 A1 B1 M1 A1 M1 A1 | (7 marks) [17 marks total] |
**Part (a)**
LM: $4mu + 3mu = 2mx + 3my$
NEL: $y - x = \frac{1}{5}u$
Solving to $y = \frac{8}{5}u$ | M1 A1 B1 M1 A1 | (5 marks)
cso
**Part (b)**
$x = \frac{11}{10}u$ or equivalent
Energy loss: $\frac{1}{2} \times 2m((2u)^2 - (\frac{11}{10}u)^2) + \frac{1}{2} \times 3m(u^2 - (\frac{8}{5}u)^2)$
$= \frac{9}{20}mu^2$ | B1 M1 A(2,1,0) A1 | (5 marks)
**Part (c)**
LM: $\frac{24}{5}mu = 3ms + mt$
NEL: $t - s = \frac{8}{5}eu$
Solving to $s = \frac{8}{5}u(3 - e)$
For a further collision: $\frac{11}{10}u > \frac{8}{5}u(3 - e)$
$e > \frac{1}{4}$ ignore $e \leq 1$ | M1 A1 B1 M1 A1 M1 A1 | (7 marks) [17 marks total]A particle $P$ of mass $2m$ is moving with speed $2u$ in a straight line on a smooth horizontal plane. A particle $Q$ of mass $3m$ is moving with speed $u$ in the same direction as $P$. The particles collide directly. The coefficient of restitution between $P$ and $Q$ is $\frac{1}{3}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $Q$ immediately after the collision is $\frac{3}{2}u$. [5]
\item Find the total kinetic energy lost in the collision. [5]
\end{enumerate}
After the collision between $P$ and $Q$, the particle $Q$ collides directly with a particle $R$ of mass $m$ which is at rest on the plane. The coefficient of restitution between $Q$ and $R$ is $e$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Calculate the range of values of $e$ for which there will be a second collision between $P$ and $Q$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2008 Q7 [17]}}