Successive oblique impactsEdexcel A-Level Further Maths: Mind map

Exam questions on Successive oblique impacts

1. A smooth particle of mass $0.5$ kg moves on a smooth horizontal table. Two fixed smooth vertical walls $W_1$ and $W_2$ meet at right angles: $W_1$ is parallel to $\mathbf{i}$ and $W_2$ is parallel to $\mathbf{j}$. The particle has velocity $(8\mathbf{i}+6\mathbf{j})$ m s$^{-1}$, hits $W_1$ and then hits $W_2$. The coefficient of restitution between the particle and each wall is $0.5$.
   Find the total kinetic energy lost by the particle in the two impacts.2 marks
2. A smooth sphere of mass $0.2$ kg moves on a smooth horizontal surface between two fixed parallel smooth vertical walls $W_1$ and $W_2$. It hits $W_1$ with speed $12$ m s$^{-1}$, its direction of motion making an angle of $60^\circ$ with the wall, and then crosses to hit $W_2$. The coefficient of restitution between the sphere and each wall is $\frac23$.
   Find the total kinetic energy lost by the sphere in the two impacts.2 marks
3. A smooth particle $P$ of mass $0.4$ kg moves on a smooth horizontal surface with velocity $(6\mathbf{i}+8\mathbf{j})$ m s$^{-1}$. It hits a fixed smooth vertical wall $W_1$ that is parallel to $\mathbf{j}$, and then hits a fixed smooth vertical wall $W_2$ that is parallel to $\mathbf{i}$. The coefficient of restitution between $P$ and $W_1$ is $\frac13$, and between $P$ and $W_2$ is $\frac12$.
   Find the speed of $P$ immediately after it hits $W_1$, and the angle between its path and $W_1$.3 marks