A roller-coaster car has mass $100 \mathrm{kg}$. It carries two passengers, each of mass between $50 \mathrm{kg}$ and $80 \mathrm{kg}$. The car becomes detached from the drive chain and continues to travel along the ride with no drive force and no braking force. The car comes to instantaneous rest at the highest point of the ride and then descends under gravity to reach the lowest point of the ride. The highest point is $12 \mathrm{m}$ vertically above the lowest point. The car travels $100 \mathrm{m}$ along the track while descending through $12 \mathrm{m}$. When the car passes through the lowest point it has speed $15 \mathrm{m}{ \mathrm{s}}^{-1}$.

Show that the average frictional force is less than $20 \mathrm{N}$.

A roller-coaster car has mass $100 \mathrm{kg}$. It carries two passengers, each of mass between $50 \mathrm{kg}$ and $80 \mathrm{kg}$. The car becomes detached from the drive chain and continues to travel along the ride with no drive force and no braking force. The car comes to instantaneous rest at the highest point of the ride and then descends under gravity to reach the lowest point of the ride. The highest point is $12 \mathrm{m}$ vertically above the lowest point. The car travels $100 \mathrm{m}$ along the track while descending through $12 \mathrm{m}$. When the car passes through the lowest point it has speed $15 \mathrm{m}{ \mathrm{s}}^{-1}$.

If no other non-gravitational resistances act, show that the average frictional force must be at least $15 \mathrm{N}$.

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$.

Find the gain in the gravitational potential energy of the ball in rising to $\theta =120°$. (Use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$.

Show that the speed of the ball at this position is $\sqrt{v^2-30}{ \mathrm{ms}}^{-1}$. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$.

In the first case to be considered, $v=8$. Find the speed of the ball when $\theta =120°$. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$. Now the ball comes to rest when $\theta =120°$. What was its initial speed, $v?$ (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$. Now consider, $v=3.5$. What is the value of $\theta$ when the ball comes to instantaneous rest?

A ball, of mass $1 \mathrm{kg}$, moves in an arc of a vertical circle of radius $1 \mathrm{m}$ by rotating on the end of a light rod. Air resistance can be ignored. Initially the rod hangs vertically. The ball is then given an initial horizontal speed of $v \mathrm{m}{ \mathrm{s}}^{-1}$. It travels in a circular arc through an angle $\theta$.

In the final case to be considered, the ball is just able to make a complete circle (so its speed at the top of the circular path is $0{ \mathrm{ms}}^{-1}$). What was its initial speed, $v?$ (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)