| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics A AS (Further Mechanics A AS) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring conservation of momentum, coefficient of restitution, and elastic collision principles across sequential collisions. Part (d) requires careful analysis of velocity conditions for a second collision. While systematic, it demands more problem-solving than standard single-collision questions and involves algebraic manipulation across multiple parts, placing it moderately above average difficulty. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | Let the velocities of A and B after the first collision be v and |
| Answer | Marks | Guidance |
|---|---|---|
| 1 2 | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 1 | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | A1 | 1.1 |
| So magnitude of impulse of A on B =3×( 5.6−0 )=16.8(Ns) | A1 | 1.1 |
| Answer | Marks |
|---|---|
| (b) | KE after−KE before |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 2 | B1 | 1.1 |
| Answer | Marks |
|---|---|
| (c) | Let the velocities of B and C after the second collision be w |
| Answer | Marks | Guidance |
|---|---|---|
| B C | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| C B | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| B B | M1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| B | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | 1.225−0.78125u<−1.4 | |
| For A and B to collide again, | M1 | 3.1b |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒u>3.36 | A1cao | 2.5 |
Question 3:
3 | (a) | Let the velocities of A and B after the first collision be v and
1
v ms-1 to the right.
2
2v +3v =7×2=14
1 2 | M1 | 3.3 | COLM
v −v =7
2 1 | M1 | 3.3 | NELR
v =−1.4
1 | A1 | 1.1 | These two marks can be earned
v =5.6
2 | A1 | 1.1 | in (b) or (c) or (d)
So magnitude of impulse of A on B =3×( 5.6−0 )=16.8(Ns) | A1 | 1.1 | Condone –16.8
Award all 5 marks if 16.8 is
obtained by correct working, even
when the unused velocity is missing
(or wrong)
[5]
(b) | KE after−KE before
= 1(2)(−1.4)2 + 1(3)(5.6)2 −1(2)(7)2 =0 as required.
2 2 2 | B1 | 1.1
[1]
(c) | Let the velocities of B and C after the second collision be w
B
and w ms-1 to the right.
C
3w +5w =3×5.6−5u=16.8−5u
B C | M1 | 3.3 | Condone using same variables that
they might have used in part (a).
w −w =0.25 ( 5.6+u )
C B | M1 | 3.3
3w +5 ( w +1.4+0.25u )=16.8−5u
B B | M1 | 1.1 | Dependent on previous M1M1
⇒w =1.225−0.78125u
B | A1 | 1.1 | AG Must be convincingly reached.
[4]
(d) | 1.225−0.78125u<−1.4
For A and B to collide again, | M1 | 3.1b | Or ‘w <−1.4’
B
⇒u>3.36 | A1cao | 2.5
[2]3 Three small uniform spheres A, B and C have masses $2 \mathrm {~kg} , 3 \mathrm {~kg}$ and 5 kg respectively. The spheres move in the same straight line on a smooth horizontal table, with B between A and C . Sphere A moves towards B with speed $7 \mathrm {~m} \mathrm {~s} ^ { - 1 } , \mathrm {~B}$ is at rest and C moves towards B with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-3_181_1291_461_251}
Spheres A and B collide. Collisions between A and B can be modelled as perfectly elastic.
\begin{enumerate}[label=(\alph*)]
\item Determine the magnitude of the impulse of A on B in this collision.
\item Use this collision to verify that in a perfectly elastic collision no kinetic energy is lost.
After the collision between A and B, sphere B subsequently collides with C. The coefficient of restitution between B and C is $\frac { 1 } { 4 }$.
\item Show that, after the collision between B and C , B has a speed of $( 1.225 - 0.78125 \mathrm { u } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ towards C.
\item Determine the range of values for $u$ for there to be a second collision between A and B .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2021 Q3 [12]}}