Question 4 - A-Level Maths

OCR MEI Further Mechanics B AS 2019 June — Question 4 13 marks

Exam Board	OCR MEI
Module	Further Mechanics B AS (Further Mechanics B AS)
Year	2019
Session	June
Marks	13
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Oblique and successive collisions
Type	Successive collisions with wall rebound
Difficulty	Challenging +1.2 This is a multi-part oblique collision problem requiring resolution of velocities, conservation of momentum, Newton's law of restitution, and analysis of subsequent motion. While it involves several steps and careful component resolution (sin α = 0.6 given), the techniques are standard for Further Mechanics. Part (c) requires setting up an inequality for a second collision, which adds modest problem-solving demand beyond routine application. Overall, this is moderately above average difficulty for A-level but uses well-practiced methods from the Further Mechanics syllabus.
Spec	6.03b Conservation of momentum: 1D two particles 6.03c Momentum in 2D: vector form 6.03d Conservation in 2D: vector momentum 6.03k Newton's experimental law: direct impact 6.03l Newton's law: oblique impacts

4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-3_725_1131_1361_242} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Find the speeds of A and B immediately after the collision.
Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
How does your answer to part (b) change if the contact between B and the wall is not smooth?

Show mark scheme Show mark scheme source

Question 4:

Answer	Marks	Guidance
4	(a)	0.2×0.5cos𝛼 [+0] = 0.2𝑎 +0.5𝑏

Or could be:

Answer	Marks
0.2×0.5×0.8 = 0.2𝑎+0.5𝑏	a and b are speeds of A

and B to right parallel to

loc after collision

Answer	Marks	Guidance
𝑏−𝑎 = −𝑒(0−0.5×cos𝛼)	M1	3.3

separation reversed

Answer	Marks	Guidance
0.2𝑎 +0.5𝑏 = .08 and 𝑏−𝑎 = 0.3	A1	1.1
Solve sim equations	M1	1.1

implied by a or b correct.

Answer	Marks	Guidance
𝑎 = −0.1;𝑏 = 0.2	A1	1.1
Speed of A perp loc = 0.3	B1	1.2

Speed of A is √(0.32+0.12)

Answer	Marks	Guidance
Speed of B is √(0.32 +0.22)	M1	2.2a

Speed of A is 0.316 or √0.1 m−1

Answer	Marks	Guidance
Speed of B is 0.361 or √0.13 m−1	A1	1.1

Allow 2s.f. or better

[8]

Answer	Marks	Guidance
(b)	Yes because vel perp loc is same (0.3) for both
A and B	B1	2.4
acceptable	Vertical speed the same is

ok for B1

[1]

Answer	Marks	Guidance
(c)	After hitting wall speed of B along loc must be
greater than 0.1 m−1 towards A	M1	3.1b

Use e = speed of B after / speed of B before

Answer	Marks	Guidance
collision with wall	M1	3.4

equal to e"b"

Answer	Marks	Guidance
[e] > 0.5	A1	2.5

also be present. Allow

both 0.5e1 and

0.5e1

[3]

Answer	Marks
(d)	B would not have same speed as A perp loc so

after collision with the wall so there would not

Answer	Marks	Guidance
be a second collision between A and B	B1	3.5a

[1]

Question 4:
4 | (a) | 0.2×0.5cos𝛼 [+0] = 0.2𝑎 +0.5𝑏 | M1 | 3.3 | CoM
Or could be:
0.2×0.5×0.8 = 0.2𝑎+0.5𝑏 | a and b are speeds of A
and B to right parallel to
loc after collision
𝑏−𝑎 = −𝑒(0−0.5×cos𝛼) | M1 | 3.3 | NEL | M1 if speed approach /
separation reversed
0.2𝑎 +0.5𝑏 = .08 and 𝑏−𝑎 = 0.3 | A1 | 1.1
Solve sim equations | M1 | 1.1 | Get eqn in 1 variable; may be
implied by a or b correct.
𝑎 = −0.1;𝑏 = 0.2 | A1 | 1.1
Speed of A perp loc = 0.3 | B1 | 1.2 | Allow 0.5sin
Speed of A is √(0.32+0.12)
Speed of B is √(0.32 +0.22) | M1 | 2.2a | For either
Speed of A is 0.316 or √0.1 m−1
Speed of B is 0.361 or √0.13 m−1 | A1 | 1.1 | For both | AEF
Allow 2s.f. or better
[8]
(b) | Yes because vel perp loc is same (0.3) for both
A and B | B1 | 2.4 | “speed parallel to the wall” also
acceptable | Vertical speed the same is
ok for B1
[1]
(c) | After hitting wall speed of B along loc must be
greater than 0.1 m−1 towards A | M1 | 3.1b
Use e = speed of B after / speed of B before
collision with wall | M1 | 3.4 | Setting new horizontal speed
equal to e"b"
[e] > 0.5 | A1 | 2.5 | Must have strict inequality | Upper limit of e might
also be present. Allow
both 0.5e1 and
0.5e1
[3]
(d) | B would not have same speed as A perp loc so
after collision with the wall so there would not
be a second collision between A and B | B1 | 3.5a | No collision is enough for B1
[1]

Show LaTeX source

4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.\\
Immediately before the collision, A is moving with speed $0.5 \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ with the line of centres, where $\sin \alpha = 0.6$, and B is moving with speed $0.3 \mathrm {~ms} ^ { - 1 }$ at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-3_725_1131_1361_242}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Find the speeds of A and B immediately after the collision.
\item Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
\item Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
\item How does your answer to part (b) change if the contact between B and the wall is not smooth?
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics B AS 2019 Q4 [13]}}

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