| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics A AS (Further Mechanics A AS) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Multiple wall bounces or returns |
| Difficulty | Standard +0.8 This is a multi-part mechanics question requiring understanding of coefficient of restitution, multiple collisions, and tracking two spheres simultaneously. Part (a) tests conceptual understanding, part (b) requires applying restitution formula twice, and part (c) involves complex kinematics with multiple wall bounces and sphere interactions—significantly above average difficulty but uses standard Further Mechanics techniques. |
| Spec | 6.03f Impulse-momentum: relation6.03g Impulse in 2D: vector form6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | If it were inelastic, the sphere would stick to the wall (but it |
| doesn’t). | B1 | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | Let the coefficient of restitution be e. | |
| 5 e 2 = 3 .2 | M1 | 1.1 |
| e = 0 .8 | A1 | 1.1 |
| Answer | Marks |
|---|---|
| (c) | A and B take 5 and 53 seconds respectively to reach their |
| Answer | Marks | Guidance |
|---|---|---|
| walls. | M1 | 3.1b |
| Answer | Marks | Guidance |
|---|---|---|
| So B travels an additional 52 − 53 0 .3 0 .8 = 0 .2 m | M1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | A1 ft | 1.1 |
| Answer | Marks |
|---|---|
| 0 .2 0 .2 e 0 .3 0 .3 e | 0 .5 d 0 .5 1 − d |
| Answer | Marks |
|---|---|
| 0 .2 0 .2 e 0 .3 0 .3 e | M1 |
| M1 | For 0.5/0.2 and 0.5/0.3; OR attempt |
| Answer | Marks | Guidance |
|---|---|---|
| d = 0 .3 2 | A1 ft | FT is 0.4 ˗ 0.1e |
Question 5:
5 | (a) | If it were inelastic, the sphere would stick to the wall (but it
doesn’t). | B1 | 2.4
[1]
(b) | Let the coefficient of restitution be e.
5 e 2 = 3 .2 | M1 | 1.1 | Or v/5 = 3.2/v (= e) (v = 4)
e = 0 .8 | A1 | 1.1
[2]
(c) | A and B take 5 and 53 seconds respectively to reach their
2
walls. | M1 | 3.1b
( )
So B travels an additional 52 − 53 0 .3 0 .8 = 0 .2 m | M1 | 1.1 | Using their e, but M0 for 0.3/e
The remaining 0.8 m is covered in the ratio 2 : 3 so spheres
meet 20.8=0.32 m from left-hand wall.
5 | A1 ft | 1.1 | FT is 0.4 ˗ 0.1e
Alternatively, let d m be the distance from the left-hand wall
at which A and B meet again.
0 .5 d 0 .5 1 − d
+ = +
0 .2 0 .2 e 0 .3 0 .3 e | 0 .5 d 0 .5 1 − d
+ = +
0 .2 0 .2 e 0 .3 0 .3 e | M1
M1 | For 0.5/0.2 and 0.5/0.3; OR attempt
at expression representing time to
impact for either A or B.
For correct equation involving d
and their e.
Or other
complete
method for d
d = 0 .3 2 | A1 ft | FT is 0.4 ˗ 0.1e
[3]
M1
M1
For 0.5/0.2 and 0.5/0.3; OR attempt
at expression representing time to
impact for either A or B.
For correct equation involving d
and their e.5 Fig. 5.1 shows a small smooth sphere A at rest on a smooth horizontal surface. At both ends of the surface is a smooth vertical wall.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_97_1307_351_242}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{center}
\end{figure}
Sphere A is projected directly towards the left-hand wall at a speed of $5 \mathrm {~ms} ^ { - 1 }$. Sphere A collides directly with the left-hand wall, rebounds, then collides directly with the right-hand wall. After this second collision A has a speed of $3.2 \mathrm {~ms} ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Explain how it can be deduced that the collision between A and the left-hand wall was not inelastic.
The coefficient of restitution between A and each wall is $e$.
\item Calculate the value of $e$.
Sphere A is now brought to rest and a second identical sphere B is placed on the surface. The surface is 1 m long, and A and B are positioned so that they are both 0.5 m from each wall, as shown in Fig. 5.2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_241_1307_1322_242}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\end{center}
\end{figure}
Sphere A is projected directly towards the left-hand wall at a speed of $0.2 \mathrm {~ms} ^ { - 1 }$. At the same time, B is projected directly towards the right-hand wall at a speed of $0.3 \mathrm {~ms} ^ { - 1 }$. You may assume that the duration of impact of a sphere and a wall is negligible.
\item Calculate the distance of A and B from the left-hand wall when they meet again.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2022 Q5 [6]}}