CAIE A-Level Further Math AS 3.6.2 Momentum and Impacts Questions

$$\begin{aligned} \text { PCLM: } m v=-m u \cos 60^{\circ}+2 m u \cos \theta v=-\frac{1}{2} u+2 u \cos \theta=v=\frac{1}{2} u(4 \cos \theta-1) \end{aligned}$$
First line must be seen.

AG

1

KE of $A=\frac{1}{2} m\left(v^{2}+\left(u \sin 60^{\circ}\right)^{2}\right)$
KE of $A=2 \times \mathrm{KE}$ of B, so
$$\frac{1}{2} m\left(v^{2}+\left(u \sin 60^{\circ}\right)^{2}\right)=2 \times \frac{1}{2} \times 2 m(u \sin \theta)^{2}$$$$\begin{aligned} \left(\frac{1}{2} u(4 \cos \theta-1)\right)^{2}+\frac{3}{4} u^{2}=4 u^{2}(\sin \theta)^{2} 8 \cos ^{2} \theta-2 \cos \theta-3=0 \end{aligned}$$
Use result of (a) and rearrange.
Obtain 3-term quadratic.

$\cos \theta=\frac{3}{4}, \quad-\frac{1}{2}$ but angle is acute, so $\cos \theta=\frac{3}{4}$
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