| Exam Board | OCR |
|---|---|
| Module | Further Mechanics (Further Mechanics) |
| Year | 2022 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring multiple conservation principles (momentum in two directions, restitution), geometric reasoning about collision angles, and careful tracking through multiple collision events. However, it's highly structured with guided parts leading to specific results, making it more accessible than open-ended proof questions. The oblique collision setup and symmetry conditions require solid understanding but follow standard FM techniques. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03d Conservation in 2D: vector momentum6.03i 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 |
|---|---|---|
| 8 | (a) | Because if the collision where inelastic then the |
| Answer | Marks |
|---|---|
| with B again (which we know it does). | B1 |
| [1] | 2.4 |
| (b) | Recollide at same point => v = 0 |
| Answer | Marks |
|---|---|
| u s i n m | B1 |
| Answer | Marks |
|---|---|
| [4] | 3.1b |
| Answer | Marks |
|---|---|
| 1.1 | Seen in solution |
| Answer | Marks |
|---|---|
| B | If v is used must be consistent in |
| Answer | Marks |
|---|---|
| (c) | Velocity of A in y direction unchanged |
| Answer | Marks |
|---|---|
| 5 e 3 | B1 |
| Answer | Marks |
|---|---|
| [7] | 3.4 |
| Answer | Marks |
|---|---|
| 3.2a | v = ucos |
| Answer | Marks |
|---|---|
| root | FT if sin and cos confusion in part b |
| Answer | Marks |
|---|---|
| (d) | In practice the plane cannot be smooth |
| Answer | Marks | Guidance |
|---|---|---|
| or NEL is only an approximation | B1 | |
| [1] | 3.5b | Any sensible limitation |
Question 8:
8 | (a) | Because if the collision where inelastic then the
velocity of A after the collision would be 0 (ie A
would ‘stick to the wall’) and so it would not collide
with B again (which we know it does). | B1
[1] | 2.4
(b) | Recollide at same point => v = 0
Ax
Com: usin = mv
B
v
NEL: e B =
u s i n
u s i n
m 1
=> e = =
u s i n m | B1
M1
M1
A1
[4] | 3.1b
3.3
3.3
1.1 | Seen in solution
or v + mv on RHS
Ax B
v v −
or e Bu A x =
s i n
AG
(v = eusin)
B | If v is used must be consistent in
Ax
both equations
(c) | Velocity of A in y direction unchanged
Either A’s velocity multiplied by (–)e or B’s
velocity multiplied by (–)(5/9)e after collision
with wall.
d d d d
+ or +
u c o s e u c o s e u s i n 5 e 2 u s i n
9
d d d d
+ = +
u c o s e u c o s e u s i n 5 e 2 u s i n
9
5 t a n 5 92
5 t a n + = +
e e e
2 5 2 5
2 e e 5 e 9 0 + − − =
2 2
2 5 2 e 1 e 5 1 8 0 + − =
6 3
(5e + 6)(5e – 3) = 0 => e = − o r
5 5
3 1 5
0 e 1 e = m = =
5 e 3 | B1
B1
M1
M1
M1
A1
A1
[7] | 3.4
3.4
3.1b
2.1
1.1
1.1
3.2a | v = ucos
Ay
Ignore inconsistency between
velocity and speed here.
Calculating time of travel for
either A or B from point of
collision to wall and back
Equating expressions that are in
terms of theta, e or m and a single
velocity
Reducing equation to 3 term
quadratic in e or m
Could be BC. –1.2 or 0.6
AG. Correctly rejecting negative
root | FT if sin and cos confusion in part b
Must be the correct component
Could see eg d – r
− 1 5 2 0 2 5
5 0
(d) | In practice the plane cannot be smooth
or the discs may not be the same size/radius
or air resistance may be significant
or NEL is only an approximation | B1
[1] | 3.5b | Any sensible limitation
PMT
Need to get in touch?
If you ever have any questions about OCR qualifications or services (including administration, logistics and teaching) please feel free to get in
touch with our customer support centre.
Call us on
01223 553998
Alternatively, you can email us on
support@ocr.org.uk
For more information visit
ocr.org.uk/qualifications/resource-finder
ocr.org.uk
Twitter/ocrexams
/ocrexams
/company/ocr
/ocrexams
OCR is part of Cambridge University Press & Assessment, a department of the University of Cambridge.
For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored. © OCR
2022 Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee. Registered in England. Registered office
The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA.
Registered company number 3484466. OCR is an exempt charity.
OCR operates academic and vocational qualifications regulated by Ofqual, Qualifications Wales and CCEA as listed in their
qualifications registers including A Levels, GCSEs, Cambridge Technicals and Cambridge Nationals.
OCR provides resources to help you deliver our qualifications. These resources do not represent any particular teaching method
we expect you to use. We update our resources regularly and aim to make sure content is accurate but please check the OCR
website so that you have the most up-to-date version. OCR cannot be held responsible for any errors or omissions in these
resources.
Though we make every effort to check our resources, there may be contradictions between published support and the
specification, so it is important that you always use information in the latest specification. We indicate any specification changes
within the document itself, change the version number and provide a summary of the changes. If you do notice a discrepancy
between the specification and a resource, please contact us.
Whether you already offer OCR qualifications, are new to OCR or are thinking about switching, you can request more
information using our Expression of Interest form.
Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.8 Two smooth circular discs, $A$ and $B$, have equal radii and are free to move on a smooth horizontal plane. The masses of $A$ and $B$ are 1 kg and $m \mathrm {~kg}$ respectively. $B$ is initially placed at rest with its centre at the origin, $O$. $A$ is projected towards $B$ with a velocity of $u \mathrm {~ms} ^ { - 1 }$ at an angle of $\theta$ to the negative $y$-axis where $\tan \theta = \frac { 5 } { 2 }$. At the instant of collision the line joining their centres lies on the $x$-axis.
There are two straight vertical walls on the plane. One is perpendicular to the $x$-axis and the other is perpendicular to the $y$-axis. The walls are an equal distance from $O$ (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-7_944_1241_694_242}
After $A$ and $B$ have collided with each other, each of them goes on to collide with a wall. Each then rebounds and they collide again at the same place as their first collision, with disc $B$ again at $O$.
The coefficient of restitution between $A$ and $B$ is denoted by $e$. The coefficient of restitution between $A$ and the wall that it collides with is also $e$ while the coefficient of restitution between $B$ and the wall that it collides with is $\frac { 5 } { 9 } e$.
It is assumed that any resistance to the motion of $A$ and $B$ may be ignored.
\begin{enumerate}[label=(\alph*)]
\item Explain why it must be the case that the collision between $A$ and the wall that it collides with is not inelastic.
\item Show that $\mathrm { e } = \frac { 1 } { \mathrm {~m} }$.
\item Show that $m = \frac { 5 } { 3 }$.
\item State one limitation of the model used.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Mechanics 2022 Q8 [13]}}