Standard +0.8 This is a mechanics problem requiring application of coefficient of restitution in 2D, conservation of momentum parallel to the barrier, and solving a trigonometric equation involving both components. It requires careful decomposition of velocities and algebraic manipulation beyond routine textbook exercises, but follows standard collision mechanics principles without requiring novel insight.
\includegraphics{figure_2}
A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4\text{ m s}^{-1}\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(0.4\). Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\). [5]
Find speed component along barrier: V cos β = 4 cos α B1
Find speed component normal to barrier: V sin β = 0⋅4 × 4 sin α B1
Find β by eliminating α with V = 2: V 2 = 2 2 = 1⋅6 2 sin 2 α + 16 cos 2 α M1
1 – sin 2 α + 0⋅16 sin 2 α = 0⋅25
0.75 25
sin 2 α = = = 0⋅8929
0.84 28
3
or cos 2 α = = 0⋅1071
28
Answer
Marks
Guidance
α = 1⋅24 rad or 70⋅9° M1 A1
5
[5]
Question 2:
2 | Find speed component along barrier: V cos β = 4 cos α B1
Find speed component normal to barrier: V sin β = 0⋅4 × 4 sin α B1
Find β by eliminating α with V = 2: V 2 = 2 2 = 1⋅6 2 sin 2 α + 16 cos 2 α M1
1 – sin 2 α + 0⋅16 sin 2 α = 0⋅25
0.75 25
sin 2 α = = = 0⋅8929
0.84 28
3
or cos 2 α = = 0⋅1071
28
α = 1⋅24 rad or 70⋅9° M1 A1 | 5 | [5]
\includegraphics{figure_2}
A small smooth ball $P$ is moving on a smooth horizontal plane with speed $4\text{ m s}^{-1}$. It strikes a smooth vertical barrier at an angle $\alpha$ (see diagram). The coefficient of restitution between $P$ and the barrier is $0.4$. Given that the speed of $P$ is halved as a result of the collision, find the value of $\alpha$. [5]
\hfill \mbox{\textit{CAIE FP2 2014 Q2 [5]}}