| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Challenging +1.2 This is a standard M3 oblique collision problem requiring resolution of velocities along/perpendicular to the line of centres, conservation of momentum in the line of centres, and Newton's restitution equation. While it involves multiple steps and careful bookkeeping with given trigonometric values, it follows a well-established procedure taught explicitly in M3 with no novel insight required. The 11 marks reflect computational length rather than conceptual difficulty, placing it moderately above average difficulty. |
| Spec | 6.03c Momentum in 2D: vector form6.03d Conservation in 2D: vector momentum6.03k Newton's experimental law: direct impact |
I don't see any mark scheme content in your message to clean up. You've provided the question number "Question 5:" and the number "5", but there's no actual marking scheme with marking annotations (M1, A1, B1, DM1, etc) or content that needs to be converted from unicode to LaTeX notation.
Please provide the complete mark scheme content you'd like me to format.5 Two smooth spheres, $A$ and $B$, have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, $A$ has velocity $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at an angle $\alpha$ to the line of centres of the spheres, and $B$ has velocity $2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at an angle $\beta$ to the line of centres, as shown in the diagram.\\
\begin{tikzpicture}[
>=Straight Barb, % Use a simple V-shaped arrowhead to match the image
velocity/.style={
postaction={decorate},
decoration={
markings,
mark=at position 0.55 with {\arrow{>}}
}
}
]
% Define the centers of the two circles
\coordinate (A) at (0,0);
\coordinate (B) at (2,0);
% Draw the dashed line of centres
\draw[densely dashed] (-2.2, 0) -- (4.2, 0) node[right, text=black] {\textsf{Line of centres}};
% Draw the two touching circles
\draw (A) circle (1);
\draw (B) circle (1);
% Add labels A and B below the circles
\node at (0, -1.4) {$A$};
\node at (2, -1.4) {$B$};
% --- Particle A ---
\def\angleA{115} % Angle for velocity vector A
% Draw velocity vector A
\draw[velocity] (\angleA:2.8) -- (A);
\node[right, xshift=2pt] at (\angleA:2.2) {$4\mathrm{\,m\,s^{-1}}$};
% Draw angle arc and label for alpha
\draw (180:0.5) arc (180:\angleA:0.5);
\node at (147.5:0.75) {$\alpha$};
% --- Particle B ---
\def\angleB{100} % Steeper angle for velocity vector B
% Draw velocity vector B
\draw[velocity] ($(B) + (\angleB:2.8)$) -- (B);
\node[right, xshift=2pt] at ($(B) + (\angleB:2.2)$) {$2.6\mathrm{\,m\,s^{-1}}$};
% Draw angle arc and label for beta
\draw ($(B) + (180:0.5)$) arc (180:\angleB:0.5);
\node at ($(B) + (140:0.75)$) {$\beta$};
\end{tikzpicture}
The coefficient of restitution between $A$ and $B$ is $\frac { 4 } { 7 }$.\\
Given that $\sin \alpha = \frac { 4 } { 5 }$ and $\sin \beta = \frac { 12 } { 13 }$, find the speeds of $A$ and $B$ immediately after the collision.\\[0pt]
[11 marks]
\hfill \mbox{\textit{AQA M3 2015 Q5 [11]}}