| Exam Board | Edexcel |
|---|---|
| Module | FM1 (Further Mechanics 1) |
| Year | 2024 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.3 This is a standard Further Mechanics 1 collision problem with straightforward application of conservation of momentum and Newton's restitution law. Part (a) is a routine 'show that' requiring two equations, part (b) requires comparing velocities (standard technique), and parts (c)-(d) are direct applications of impulse and energy formulas. While it's a multi-part question requiring careful algebra, it involves no novel insight—just systematic application of well-practiced methods from the FM1 syllabus. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03e Impulse: by a force6.03f Impulse-momentum: relation6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| CLM applied | M1 | |
| \(2m \times 3u = 2mv_A + mv_B \Rightarrow (6u = 2v_A + v_B)\) or \(2m \times 3u = -2mv_A + mv_B \Rightarrow (6u = -2v_A + v_B)\) | A1 | |
| Impact Law applied | M1 | |
| \(3ue = -v_A + v_B\) or \(3ue = v_A + v_B\) | A1 | |
| Solve for \(v_B\) | M1 | |
| \(v_B = 2u(1+e)\) | A1* | Given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solve for \(v_A\) | M1 | |
| \(v_A = u(2-e)\) or \(v_A = u(e-2)\) | A1 | |
| Complete and correct explanation: \(0 \leq e \leq 1 \Rightarrow v_A > 0 \Rightarrow A\) continues to move towards the wall \(\Rightarrow A\) will collide again with \(B\) or \(0 \leq e \leq 1 \Rightarrow v_A < 0 \Rightarrow A\) continues to move towards the wall \(\Rightarrow A\) will collide again with \(B\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rebound speed or velocity of \(B = \pm\frac{1}{2} \times 2u(1+e)\) | B1 | |
| \(\pm m[-u(1+e) - 2u(1+e)]\) | M1 | |
| \(3(1+e)mu\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt at KE loss of \(B\) | M1 | |
| \(\frac{1}{2}m\left[(2u(1+e))^2 - (u(1+e))^2\right]\) | A1 | |
| \(\frac{3mu^2(1+e)^2}{2}\) | A1 |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| CLM applied | M1 | |
| $2m \times 3u = 2mv_A + mv_B \Rightarrow (6u = 2v_A + v_B)$ **or** $2m \times 3u = -2mv_A + mv_B \Rightarrow (6u = -2v_A + v_B)$ | A1 | |
| Impact Law applied | M1 | |
| $3ue = -v_A + v_B$ **or** $3ue = v_A + v_B$ | A1 | |
| Solve for $v_B$ | M1 | |
| $v_B = 2u(1+e)$ | A1* | Given answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solve for $v_A$ | M1 | |
| $v_A = u(2-e)$ **or** $v_A = u(e-2)$ | A1 | |
| Complete and correct explanation: $0 \leq e \leq 1 \Rightarrow v_A > 0 \Rightarrow A$ continues to move towards the wall $\Rightarrow A$ will collide again with $B$ **or** $0 \leq e \leq 1 \Rightarrow v_A < 0 \Rightarrow A$ continues to move towards the wall $\Rightarrow A$ will collide again with $B$ | A1 | |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rebound speed or velocity of $B = \pm\frac{1}{2} \times 2u(1+e)$ | B1 | |
| $\pm m[-u(1+e) - 2u(1+e)]$ | M1 | |
| $3(1+e)mu$ | A1 | |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt at KE loss of $B$ | M1 | |
| $\frac{1}{2}m\left[(2u(1+e))^2 - (u(1+e))^2\right]$ | A1 | |
| $\frac{3mu^2(1+e)^2}{2}$ | A1 | |\begin{enumerate}
\item A particle $A$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal plane. Particle $A$ collides directly with a particle $B$ of mass $m$ which is at rest on the plane.
\end{enumerate}
The coefficient of restitution between $A$ and $B$ is $e$, where $e > 0$\\
(a) Show that the speed of $B$ immediately after the collision is $2 u ( 1 + e )$.
After the collision, $B$ hits a smooth fixed vertical wall which is perpendicular to the direction of motion of $B$.\\
(b) Show that there will be a second collision between $A$ and $B$.
The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$\\
Find, in simplified form, in terms of $m$, $u$ and $e$,\\
(c) the magnitude of the impulse received by $B$ in its collision with the wall,\\
(d) the loss in kinetic energy of $B$ due to its collision with the wall.
\hfill \mbox{\textit{Edexcel FM1 2024 Q4 [15]}}