| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2017 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.8 This M2 collision problem requires applying both conservation of momentum and Newton's restitution law with careful sign conventions for reversed directions, then analyzing conditions for a second collision involving a wall rebound. Part (a) involves simultaneous equations with algebraic manipulation, while part (b) requires inequality reasoning about relative velocities—more demanding than standard single-collision questions but follows established M2 patterns. |
| 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: \(-2mv + 4.5mv = 2mu_P - 3mu_Q\) | M1 | Need all four terms and dimensionally correct. Condone sign errors |
| \(2.5v = 2u_P - 3v_Q\) | A1 | Correct unsimplified equation |
| Impact law: \(\frac{1.5v + v}{u_P + u_Q} = \frac{1}{5}\) | M1 | Must be used right way round. Accept with \(e\) not substituted |
| \(12.5v = u_P + u_Q\) | A1ft | Correct unsimplified. Signs consistent with their CLM equation |
| Solve for \(u_P\) and \(u_Q\) | DM1 | Dependent on both previous M marks |
| A1 | One correct | |
| \(u_P = 8v\) and \(u_Q = 4.5v\) | A1 | Both correct — from cwo. Mark final answer — do not ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| After impact with wall: \(1.5ve\) | B1 | |
| For Q to catch P: \(1.5ve > v\) | M1 | Correct inequality for their \(1.5ve\) (could be implied by correct answer) |
| \(1 \geq e > \frac{2}{3}\) | A1 | Need both ends |
# Question 5:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| CLM: $-2mv + 4.5mv = 2mu_P - 3mu_Q$ | M1 | Need all four terms and dimensionally correct. Condone sign errors |
| $2.5v = 2u_P - 3v_Q$ | A1 | Correct unsimplified equation |
| Impact law: $\frac{1.5v + v}{u_P + u_Q} = \frac{1}{5}$ | M1 | Must be used right way round. Accept with $e$ not substituted |
| $12.5v = u_P + u_Q$ | A1ft | Correct unsimplified. Signs consistent with their CLM equation |
| Solve for $u_P$ and $u_Q$ | DM1 | Dependent on both previous M marks |
| | A1 | One correct |
| $u_P = 8v$ and $u_Q = 4.5v$ | A1 | Both correct — from cwo. Mark final answer — do not ISW |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| After impact with wall: $1.5ve$ | B1 | |
| For Q to catch P: $1.5ve > v$ | M1 | Correct inequality for their $1.5ve$ (could be implied by correct answer) |
| $1 \geq e > \frac{2}{3}$ | A1 | Need both ends |5. Two particles $P$ and $Q$, of masses $2 m$ and $3 m$ respectively, are moving in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly and, as a result of the collision, the direction of motion of $P$ is reversed and the direction of motion of $Q$ is reversed. Immediately after the collision, the speed of $P$ is $v$ and the speed of $Q$ is $\frac { 3 v } { 2 }$. The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 5 }$.
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{enumerate}[label=(\roman*)]
\item the speed of $P$ immediately before the collision,
\item the speed of $Q$ immediately before the collision.
After the collision with $P$, the particle $Q$ moves on the plane and strikes at right angles a fixed smooth vertical wall and rebounds. The coefficient of restitution between $Q$ and the wall is $e$. Given that there is a further collision between the particles,
\end{enumerate}\item find the range of possible values of $e$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2017 Q5 [10]}}