| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2015 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Standard +0.8 This is a challenging M2 mechanics problem requiring sequential collision analysis with three particles of different masses, application of both momentum conservation and restitution equations, algebraic manipulation to find velocity expressions in terms of e, determining conditions for motion reversal, and proving a second collision occurs. The multi-stage reasoning and need to track multiple velocities across collisions elevates this above standard single-collision problems. |
| 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 | Mark | Guidance |
| CLM: \(3mu = mv + 2mw\) \((3u = v + 2w)\) | M1 | Requires all three terms, but condone sign errors |
| A1 | ||
| Impact: \(w - v = 3eu\) | M1 | Impact law applied the right way round, but condone sign errors |
| A1 | A0 if signs in the two equations are not consistent | |
| \(3w = 3u(1+e)\), \(w = u(1+e)\) | DM1 | Dependent on the 2 previous M marks. Solve for \(w\) or \(v\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v = w - 3eu = u - 2ue\), speed \(= \ | u(1-2e)\ | \) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Change in direction \(1 - 2e < 0\) | M1 | Correct inequality for reversal of direction (for their \(v\)) |
| \((1 \geq)\, e > \frac{1}{2}\) | A1[2] | CWO. \(e \leq 1\) not required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v = -\frac{u}{3}\) and \(w = \frac{5}{3}u\) | B1 | |
| CLM: \(2m \times w - 3mu = 2mq + 3mr\) and \(r - q = e(w + u)\left(= \frac{16u}{9}\right)\) | M1 | CLM & impact equations |
| A1 | Both correct | |
| \(\frac{u}{3} = 2q + 3r\), \(3r - 3q = \frac{16u}{3}\) | DM1 | Dependent on previous M1. Solve for \(q\) |
| \(5q = -5u\), \(q = -u\) | A1 | |
| \(u > \frac{u}{3}\) therefore \(Q\) will collide with \(P\) a second time | A1[6] | Given answer |
## Question 7(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| CLM: $3mu = mv + 2mw$ $(3u = v + 2w)$ | M1 | Requires all three terms, but condone sign errors |
| | A1 | |
| Impact: $w - v = 3eu$ | M1 | Impact law applied the right way round, but condone sign errors |
| | A1 | A0 if signs in the two equations are not consistent |
| $3w = 3u(1+e)$, $w = u(1+e)$ | DM1 | Dependent on the 2 previous M marks. Solve for $w$ or $v$ |
## Question 7(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v = w - 3eu = u - 2ue$, speed $= \|u(1-2e)\|$ | A1[6] | Both speeds correct. Must both be positive |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Change in direction $1 - 2e < 0$ | M1 | Correct inequality for reversal of direction (for their $v$) |
| $(1 \geq)\, e > \frac{1}{2}$ | A1[2] | CWO. $e \leq 1$ not required |
## Question 7(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v = -\frac{u}{3}$ and $w = \frac{5}{3}u$ | B1 | |
| CLM: $2m \times w - 3mu = 2mq + 3mr$ and $r - q = e(w + u)\left(= \frac{16u}{9}\right)$ | M1 | CLM & impact equations |
| | A1 | Both correct |
| $\frac{u}{3} = 2q + 3r$, $3r - 3q = \frac{16u}{3}$ | DM1 | Dependent on previous M1. Solve for $q$ |
| $5q = -5u$, $q = -u$ | A1 | |
| $u > \frac{u}{3}$ therefore $Q$ will collide with $P$ a second time | A1[6] | **Given answer** |7. Three particles $P , Q$ and $R$ lie at rest in a straight line on a smooth horizontal surface with $Q$ between $P$ and $R$. Particle $P$ has mass $m$, particle $Q$ has mass $2 m$ and particle $R$ has mass $3 m$. The coefficient of restitution between each pair of particles is $e$. Particle $P$ is projected towards $Q$ with speed $3 u$ and collides directly with $Q$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $u$ and $e$,
\begin{enumerate}[label=(\roman*)]
\item the speed of $Q$ immediately after the collision,
\item the speed of $P$ immediately after the collision.
\end{enumerate}\item Find the range of values of $e$ for which the direction of motion of $P$ is reversed as a result of the collision with $Q$.
Immediately after the collision between $P$ and $Q$, particle $R$ is projected towards $Q$ with speed $u$ so that $R$ and $Q$ collide directly. Given that $e = \frac { 2 } { 3 }$
\item show that there will be a second collision between $P$ and $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2015 Q7 [14]}}