In a pinball machine, ball bearings are fired up a slope inclined at $10°$ to the horizontal. After travelling $1.2 \mathrm{m}$ the ball bearings reach a curved barrier. Find the maximum initial speed of a ball bearing if it comes to rest before getting to the curved barrier.

A diver jumps from a $10 \mathrm{m}$ tall board into a swimming pool.

The diver has an initial velocity of $u \mathrm{m}{ \mathrm{s}}^{-1}$ upwards. Find his speed when he hits the water.

A diver jumps from a $10 \mathrm{m}$ tall board into a swimming pool. The diver has an initial velocity of $u \mathrm{m}{ \mathrm{s}}^{-1}$ upwards. Find his speed when he hits the water.

What modelling assumptions have been made?

A football is kicked from ground level with speed $15{ \mathrm{ms}}^{-1}$ and rises to a height of $1.45 \mathrm{m}$. Assume that air resistance is negligible.

Find the speed of the ball when it is $1 \mathrm{m}$ above the ground.

At the top of its flight the ball is travelling horizontally.

A crate of mass $M \mathrm{kg}$ sits at the bottom of a smooth slope that is inclined at an angle $\theta$ to the horizontal. A light inextensible rope is attached to the crate and passes over a smooth pulley at the top of the slope. The part of the rope between the crate and the pulley is parallel to the slope. The other end of the rope hangs vertically and at the other end there is a ball of mass $m \mathrm{kg}$. The system is released from rest and the ball reaches the ground with speed $v{ \mathrm{ms}}^{-1}$ after descending a distance of $h \mathrm{m}$.

Find expressions for the decrease in potential energy for the ball.

A crate of mass $M \mathrm{kg}$ sits at the bottom of a smooth slope that is inclined at an angle $\theta$ to the horizontal. A light inextensible rope is attached to the crate and passes over a smooth pulley at the top of the slope. The part of the rope between the crate and the pulley is parallel to the slope. The other end of the rope hangs vertically and at the other end there is a ball of mass $m \mathrm{kg}$. The system is released from rest and the ball reaches the ground with speed $v{ \mathrm{ms}}^{-1}$ after descending a distance of $h \mathrm{m}$.

Find expressions for the increase in kinetic energy for the ball.

A crate of mass $M \mathrm{kg}$ sits at the bottom of a smooth slope that is inclined at an angle $\theta$ to the horizontal. A light inextensible rope is attached to the crate and passes over a smooth pulley at the top of the slope. The part of the rope between the crate and the pulley is parallel to the slope. The other end of the rope hangs vertically and at the other end there is a ball of mass $m \mathrm{kg}$. The system is released from rest and the ball reaches the ground with speed $v{ \mathrm{ms}}^{-1}$ after descending a distance of $h \mathrm{m}$. Find expressions for the increase in mechanical energy for the crate. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A piece of sculpture includes a vertical metal circle with radius $2.45 \mathrm{m}$. A particle of mass $0.2 \mathrm{kg}$ sits at point $A$ on top of the sculpture at the top of the circle (on the outside of the circle). The particle is gently displaced and slides down the circle until it reaches point $B$, which is level with the centre of the circle. It then falls a further $3.6 \mathrm{m}$ vertically to hit the ground at point $C$.

Use the work-energy principle to find:

the speed of the particle when it reaches point $B$