\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]

Question 1:
1 | Along line of centres, PCLM: 5mv mv mucos
B A | M1 | Must include correct masses.
NEL: v v  1 ucos
B A 2 | M1 | Signs consistent with PCLM equation.
u u
v  cos, v  cos
B 4 A 4 | A1
Perpendicular to line of centres: speed of A is usin | B1
1   u  2 usin2  1  u  2
m    cos   5m  cos 
2   4   2 4 
  | M1 | Equate final kinetic energies, 3 terms, correct
masses.
cos2  4 , cos 2 , tan 1
5 5 2 | A1
6
Question | Answer | Marks | Guidance

\includegraphics{figure_1}

Two smooth uniform spheres $A$ and $B$ of equal radii have masses $m$ and $5m$ respectively. Sphere $A$ is moving on a smooth horizontal surface with speed $u$ when it collides with sphere $B$ which is at rest on the surface. Immediately before the collision, $A$'s direction of motion makes an angle of $\theta$ with the line of centres. After the collision, the kinetic energies of $A$ and $B$ are equal. The coefficient of restitution between the spheres is $\frac{1}{3}$.

Find the value of $\tan\theta$. [6]

\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q1 [6]}}