Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 9: EXERCISE 1F

An ice hockey puck slides along a rink at a constant speed of $10 \mathrm{m}{ \mathrm{s}}^{-1}.$ It strikes the boards at the edge of the rink $20 \mathrm{m}$ away and slides back along the rink at $8 \mathrm{m}{ \mathrm{s}}^{-1}$ until going into the goal $40 \mathrm{m}$ from the board. Sketch a velocity-time graph and a displacement-time graph for the motion, measuring displacement from the starting point in the original direction of motion.

A bowling ball rolls down an alley with initial speed $8 \mathrm{m}{ \mathrm{s}}^{-1}$ and decelerates at a constant rate of $0.8 \mathrm{m}{ \mathrm{s}}^{-2}.$ After $2.5 \mathrm{s}$ it strikes a pin and instantly slows down to $2 \mathrm{m}{ \mathrm{s}}^{-1}.$ It continues to decelerate at the same constant rate until coming to rest.

Sketch a velocity-time graph and a displacement-time graph for the motion.

In a game of blind cricket, a ball is rolled towards a player with a bat $20 \mathrm{m}$ away, who tries to hit the ball. On one occasion, the ball is rolled towards the batsman at a constant speed of $4 \mathrm{m}{ \mathrm{s}}^{-1}.$ The batsman hits the ball back directly where it came from with initial speed $6 \mathrm{m}{ \mathrm{s}}^{-1}$ and decelerating at a constant rate of $0.5 \mathrm{m}{ \mathrm{s}}^{-2}.$ Sketch a velocity-time graph and a displacement-time graph for the motion, taking the original starting point as the origin and the original direction of motion as positive.

A ball is dropped from rest $20 \mathrm{m}$ above the ground. It accelerates towards the ground at a constant rate of $10 \mathrm{m}{ \mathrm{s}}^{-2}.$ It bounces on the ground and leaves with a speed that is half the speed it struck the ground original. The ball is then caught when it reaches the highest point of its bounce. Sketch a velocity-time graph and a displacement-time graph for the motion, measuring displacement above the ground.

A ball is thrown towards a wall, which is $5 \mathrm{m}$ away, at $2.25 \mathrm{m}{ \mathrm{s}}^{-1}.$ It slows down at a constant rate of $0.2 \mathrm{m}{ \mathrm{s}}^{-2}$ until it strikes the wall. It bounces back at $80\%$ of the speed it struck the wall originally. It again slows down at a constant rate of $0.2 \mathrm{m}{ \mathrm{s}}^{-2}$ until coming to rest. Sketch a velocity-time graph and a displacement-time graph for the motion, measuring displacement from the wall, taking the direction away from the wall as positive.

A billiard ball is on the centre spot of a $6 \mathrm{m}$ long table and is struck towards one of the cushions with initial speed $3.1 \mathrm{m}{ \mathrm{s}}^{-1}.$ It slows on the table at $0.2 \mathrm{m}{ \mathrm{s}}^{-2}.$ When it bounces of the cushion its speed reduces to $70\%$ of the speed with which it struck the cushion. The ball is left until it comes to rest.

Sketch the velocity-time and displacement-time graphs for the ball, taking the centre of the table as the origin for displacement and the original direction of motion as positive.

A billiard ball is on the centre spot of a $6 \mathrm{m}$ long table and is struck towards one of the cushions with initial speed $3.1 \mathrm{m}{ \mathrm{s}}^{-1}.$ It slows on the table at $0.2 \mathrm{m}{ \mathrm{s}}^{-2}.$ When it bounces of the cushion its speed reduces to $70\%$ of the speed with which it struck the cushion. The ball is left until it comes to rest.

What assumptions have been made in your answer?

A ball is released from rest $20 \mathrm{m}$ above the ground and accelerates under gravity at $10 \mathrm{m}{ \mathrm{s}}^{-2}.$ When it bounces its speed halves. If bounce $n$ occurs at time $t_n$ the speed after the bounce is $v_n.$ Show that $v_n=15-2.5t_n$ and deduce that, despite infinitely many bounces, the ball stops bouncing after $6 \mathrm{s}.$