| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow w = \frac{3u}{4(k+1)}\) (*) | M1 A1, M1 A1, M1, A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Solve \(k: \Rightarrow k = \frac{1}{2}\) | M1 A1, M1 A1, M1 A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{1}{32}mu^2\) | B1, M1 A1, A1 | (4 marks) |
## Part (a)
CLM ($\leftrightarrow$): $\mu u\cos 60 = mv + kmw$
NLI: $\frac{1}{2}\mu u\cos 60 = w - v$
Solve for $w: (1+k)w = \frac{1}{2}u(1 + \frac{1}{2})$
$\Rightarrow w = \frac{3u}{4(k+1)}$ (*) | M1 A1, M1 A1, M1, A1 | (6 marks)
## Part (b)
Solve for $v: v = \frac{u(2-k)}{4(k+1)}$
$\tan\theta = 2\sqrt{3}\frac{u\sin 60}{v} = \frac{u\sqrt{3}}{2}\frac{4(k+1)}{u(2-k)}$
Solve $k: \Rightarrow k = \frac{1}{2}$ | M1 A1, M1 A1, M1 A1 | (6 marks)
## Part (c)
$k = \frac{1}{2} \Rightarrow v = \frac{u}{4}, w = \frac{u}{2}$
KE loss = $\frac{1}{2}mu^2 - (\frac{1}{2}m\cdot\frac{u^2}{16} + \frac{1}{2}m\cdot\frac{u^2}{2} + \frac{1}{4} + \frac{1}{2}\cdot 2m\cdot\frac{u^2}{4})$
$= \frac{1}{2}mu^2(1 - \frac{1}{16} - \frac{3}{4} - \frac{1}{8})$
$= \frac{1}{32}mu^2$ | B1, M1 A1, A1 | (4 marks)
**Total: 16 marks**
---Two smooth uniform spheres $A$ and $B$ have equal radii. Sphere $A$ has mass $m$ and sphere $B$ has mass $km$. The spheres are at rest on a smooth horizontal table. Sphere $A$ is then projected along the table with speed $u$ and collides with $B$. Immediately before the collision, the direction of motion of $A$ makes an angle of $60°$ with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is $\frac{1}{2}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ immediately after the collision is $\frac{3u}{4(k + 1)}$.
[6]
Immediately after the collision the direction of motion of $A$ makes an angle arctan $(2\sqrt{3})$ with the direction of motion of $B$.
\item Show that $k = \frac{1}{2}$.
[6]
\item Find the loss of kinetic energy due to the collision.
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2006 Q5 [16]}}