Ball $X$ has mass $0.03 \mathrm{kg}$. It falls vertically from rest from a window that is $30 \mathrm{m}$ above the ground. Ball $Y$ has mass $0.01 \mathrm{kg}$. At the same time that ball $X$ starts to fall, ball $Y$ is projected vertically upwards from ground level directly towards ball $X$. The initial speed of ball $Y$ is $20 \mathrm{m}{ \mathrm{s}}^{-1}$ vertically upwards.

The balls coalesce and the combined object falls to the ground.

Show that the combined object reaches the ground $2.68$ seconds after ball $X$ started to fall.

Three balls, $A, B$ and $C$, are in that order in a straight line on a smooth horizontal surface. $A$ has mass $0.4 \mathrm{kg}$ and is moving at $4 \mathrm{m}{ \mathrm{s}}^{-1}$ towards $B$. $B$ has mass $m \mathrm{kg}$ and is stationary. $C$ has mass $0.25 \mathrm{kg}$ and is moving at $0.8 \mathrm{m}{ \mathrm{s}}^{-1}$ away from $B$. $A$ hits $B$ and then $B$ hits $C$. There are no further impacts. $A$ and $C$ now each have a speed of $1 \mathrm{m}{ \mathrm{s}}^{-1}$ and are both moving in directions away from $B$. Find the range of possible values of $m$.

A ball of mass $0.6 \mathrm{kg}$ is dropped from a height of $1.8 \mathrm{m}$ onto a solid floor. Each time the ball bounces on the floor it loses $10\%$ of its speed.

Work out how much momentum was absorbed by the floor in the first bounce.

A ball of mass $0.6 \mathrm{kg}$ is dropped from a height of $1.8 \mathrm{m}$ onto a solid floor. Each time the ball bounces on the floor it loses $10\%$ of its speed.

Show that the ball first fails to reach a height of $1 \mathrm{m}$ after the third bounce.

A ball of mass $0.6 \mathrm{kg}$ is dropped from a height of $1.8 \mathrm{m}$ onto a solid floor. Each time the ball bounces on the floor it loses $10\%$ of its speed.

What modelling assumptions have you made?

Ball $X$ has mass $30 \mathrm{g}$ and is moving at $0.51 \mathrm{m}{ \mathrm{s}}^{-1}$. The direction in which $X$ is travelling is taken as the positive direction. Ball $Y$ has mass $60 \mathrm{g}$ and is stationary. Ball $X$ collides with ball $Y$ and, after the impact, ball $X$ moves at $0.01 \mathrm{m}{ \mathrm{s}}^{-1}$ in the positive direction. Ball $Y$ then hits a wall and rebounds with half the speed with which it hit the wall.

Work out how much momentum was absorbed by the wall.

Ball $X$ has mass $30 \mathrm{g}$ and is moving at $0.51 \mathrm{m}{ \mathrm{s}}^{-1}$. The direction in which $X$ is travelling is taken as the positive direction. Ball $Y$ has mass $60 \mathrm{g}$ and is stationary. Ball $X$ collides with ball $Y$ and, after the impact, ball $X$ moves at $0.01 \mathrm{m}{ \mathrm{s}}^{-1}$ in the positive direction. Ball $Y$ then hits a wall and rebounds with half the speed with which it hit the wall.

After rebounding from the wall, ball $Y$ goes on to hit ball $X$.

Explain why ball $X$ must be travelling in the negative direction after being hit by ball $Y$.

Ball $X$ has mass $30 \mathrm{g}$ and is moving at $0.51 \mathrm{m}{ \mathrm{s}}^{-1}$. The direction in which $X$ is travelling is taken as the positive direction. Ball $Y$ has mass $60 \mathrm{g}$ and is stationary. Ball $X$ collides with ball $Y$ and, after the impact, ball $X$ moves at $0.01 \mathrm{m}{ \mathrm{s}}^{-1}$ in the positive direction. Ball $Y$ then hits a wall and rebounds with half the speed with which it hit the wall.

After rebounding from the wall, ball $Y$ goes on to hit ball $X$.

After this impact ball $X$ has speed $0.15 \mathrm{m}{ \mathrm{s}}^{-1}$.

Find the final velocity of ball $Y$.

Balls $X, Y$ and $Z$ lie at rest on a smooth horizontal surface, with $Y$ between $X$ and $Z$. Balls $X$ and $Z$ each have mass $2 \mathrm{kg}$ and ball $Y$ has mass $1 \mathrm{kg}$. Ball $X$ is given a velocity of $1 \mathrm{m}{ \mathrm{s}}^{-1}$ towards ball $Y$. Balls $X$ and$Y$ collide. After this collision the speed of ball $Y$ is three times the speed of ball $X$. Ball $Y$ goes on to collide with ball $Z$. After this collision the speed of ball $Y$ is the same as the speed of ball $X$, and the speed of ball $Z$ is twice the speed of ball $Y$. Finally ball $Y$ collides with ball $X$ again. After this collision the speed of ball $Y$ is twice the speed of ball $X$, and the speed of ball $Z$ is four times the speed of ball $Y$. Show that the balls are now all travelling in the same direction and that no further collisions occur.