\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]

Question 1:
1 | 1 1 1
Along line of centres, PCLM: 2m ucosmu 2m ucosm u
2 2 4 | M1 | Masses must be included. Allow sign errors.
5
cos
8 | A1
1 1 1 
NEL: ucos ue  ucosu 
2 4 2  | M1 | Allow sign errors, e must be on correct side.
3
e
7 | A1 | AEF
4
Question | Answer | Marks | Guidance

\includegraphics{figure_1}

Two smooth uniform spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $\frac{1}{2}u$ respectively. Immediately before the collision, $A$'s direction of motion is along the line of centres, and $B$'s direction of motion makes an angle $\theta$ with the line of centres (see diagram).

As a result of the collision, the direction of motion of $A$ is reversed and its speed is reduced to $\frac{1}{4}u$. The direction of motion of $B$ again makes an angle $\theta$ with the line of centres, but on the opposite side of the line of centres. The speed of $B$ is unchanged.

Find the value of the coefficient of restitution between the spheres. [4]

\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q1 [4]}}