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 $6 \mathrm{s}.$

A man and his young son play a game. The man rolls a ball along the ground. His son runs after the ball to fetch it.

The ball starts rolling at $10 \mathrm{m}{ \mathrm{s}}^{-1}$ but decelerates at a constant rate of $2 \mathrm{m}{ \mathrm{s}}^{-2}.$ Find the distance covered when it comes to rest.

A man and his young son play a game. The man rolls a ball along the ground. His son runs after the ball to fetch it.

Once the ball has stopped, the boy runs to fetch it. He starts from rest beside his father and accelerates at a constant rate of $2 \mathrm{m}{ \mathrm{s}}^{-2}$ for $3 \mathrm{s}$ before maintaining a constant speed. Find the time taken to reach the ball.

A car is travelling at $15 \mathrm{m}{ \mathrm{s}}^{-1}$ when the speed limit increases and the car accelerates at a constant rate of $3 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching a top speed of $30 \mathrm{m}{ \mathrm{s}}^{-1}.$

Find the distance covered until reaching top speed.

A car is travelling at $15 \mathrm{m}{ \mathrm{s}}^{-1}$ when the speed limit increases and the car accelerates at a constant rate of $3 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching a top speed of $30 \mathrm{m}{ \mathrm{s}}^{-1}.$

Once the car is at top speed, there is a set of traffic lights $600 \mathrm{m}$ away. The car maintains $30 \mathrm{m}{ \mathrm{s}}^{-1}$ until it starts to decelerate at a constant rate of $5 \mathrm{m}{ \mathrm{s}}^{-2}$ to come to rest at the lights. Find the time taken from reaching top speed until it comes to rest at the traffic lights.

In a race, the lead runner is $60 \mathrm{m}$ ahead of the chaser with $200 \mathrm{m}$ to go and is running at $4 \mathrm{m}{ \mathrm{s}}^{-1}.$ The chaser is running at $5 \mathrm{m}{ \mathrm{s}}^{-1}.$

Find the minimum constant acceleration required by the chaser to catch the lead runner.

In a race, the lead runner is $60 \mathrm{m}$ ahead of the chaser with $200 \mathrm{m}$ to go and is running at $4 \mathrm{m}{ \mathrm{s}}^{-1}.$ The chaser is running at $5 \mathrm{m}{ \mathrm{s}}^{-1}.$

If the lead runner is actually accelerating at a constant rate of $0.05 \mathrm{m}{ \mathrm{s}}^{-2},$ find the minimum constant acceleration required by the chaser to catch the lead runner.

A jet aeroplane coming in to land at $100 \mathrm{m}{ \mathrm{s}}^{-1}$ needs $800 \mathrm{m}$ of runway.

On an aircraft carrier, the aeroplane has only $150 \mathrm{m}$ to stop. There are hooks on the aeroplane that catch arresting wires to slow it down. If the aeroplane catches the hook $50 \mathrm{m}$ after landing, find the deceleration during the last $100 \mathrm{m}.$