Work & Energy (Cambridge (CIE) AS Maths: Mechanics): Exam Questions

A company uses a cable to move a crate of mass 50 kg up a 20 metre ramp inclined at 40° to the horizontal, applying a force of 400 N as shown in the diagram. The crate moves along the line of greatest slope. The frictional force acting on the crate is $\mu R$ N, where $\mu$ is the coefficient of friction between the crate and the ramp.

Given that the work done against friction as the crate moves the full length of the ramp is 1500 J, find

(i) the frictional force acting on the crate,

(ii) the magnitude of the normal contact force $R$,

(iii) the coefficient of friction $\mu$, giving your answer correct to 2 decimal places.

By first finding an exact value for the component of the weight of the crate parallel to the direction of motion, find the work done against gravity.

Find the net work done by all the forces acting on the crate, including the force due to gravity.

A cyclist is travelling at a constant speed of 10 ${\text{m s}}^{-1}$ on horizontal ground when they see traffic ahead and apply the brakes.

Modelling the cyclist and their bicycle as a single particle of total mass $m$ kg, find their kinetic energy before the brakes are applied.

The cyclist slows down over 16 metres with constant deceleration, then continues at a constant speed of $u{\text{ m s}}^{-1}$.

Given that the loss in kinetic energy is $42m$ joules, find the value of $u$.

The work done by the resisting force of the brakes as the cyclist slows down is 3360 J.

(i) Find the combined mass of the cyclist and their bicycle.

(ii) State an assumption you have made in part (c)(i).

A bus of mass 9000 kg is travelling at a speed of 15 ${\text{m s}}^{-1}$ when the driver applies the brakes. The bus comes to a stop 38 metres later.

(i) Draw a diagram showing the forces acting on the bus as it comes to a stop. The bus may be modelled as a particle travelling along a straight horizontal road.

(ii) Show that the decrease in kinetic energy as the bus comes to a stop is 1012.5 kJ.

(iii) Find the magnitude of the resultant force acting on the bus.

The bus then accelerates with a driving force of $D$ N against a constant resisting force of 206 N. It reaches 15 ${\text{m s}}^{-1}$ again after travelling 300 m.

(i) Find the magnitude of $D$.

(ii) State an assumption you have made about the driving force $D$.

A car of mass 880 kg is driving down a hill inclined at 20° to the horizontal, along the line of greatest slope. As the car moves 400 metres down the hill its speed increases from 12 ${\text{m s}}^{-1}$ to 18 ${\text{m s}}^{-1}$.

Find

(i) the loss in gravitational potential energy,

(ii) the gain in kinetic energy.

State whether any external forces other than gravity did work on the car, giving a reason for your answer.

A skateboarder moves off from rest at the top of a ramp and passes through the point $P$ with a speed of 2 ${\text{m s}}^{-1}$. The point $Q$ is 4 m vertically below $P$.

Assuming that there is no resistance to motion, find the speed with which the skateboarder passes through $Q$.

The skateboarder and their equipment may be modelled as a particle of mass 60 kg.

Assuming instead that there is a constant resistance to motion, and that the work done against this resistance is 333 J, find the speed with which the skateboarder passes through $Q$.

A toboggan team consisting of two children and their sled moves down a course of varying gradients. The course is 400 m long and the vertical height of the hill is 35 m. The team starts from rest at the top and does no work as it moves down the course.

The frictional force on the sled is 8 N and all other resistances to motion may be ignored. The two children and their sled may be modelled as a single particle of mass 100 kg.

Find the speed of the team when they reach the end of the course.

A box of mass 4 kg slides down the line of greatest slope of a ramp inclined at $\alpha$° to the horizontal, with constant acceleration 2 ${\text{m s}}^{-2}$. Two markers on the ramp are 7 metres apart, and the box passes the second marker exactly 2 seconds after passing the first.

Show that the box passes the first marker with speed 1.5 ${\text{m s}}^{-1}$, and find the speed of the box as it passes the second marker.

The work done against friction on the box as it slides between the two markers is 84 J.

Find

(i) the magnitude of the frictional force,

(ii) the value of $\alpha$,

(iii) the work done by gravity during this time.

A small alien spaceship, of mass 150 kg, is flying horizontally with constant velocity $v{\text{ m s}}^{-1}$. Its engines provide a driving force of 2000 N forwards, and a lift force acts vertically upwards on the spaceship.

Draw a diagram showing the forces acting on the spaceship, including the magnitudes of the lift force and of the resistive force from air resistance.

Given that $v=200$, find the work done by each of the four forces acting on the spaceship over a ten second period.

The alien pilot spots a field that would be perfect for a crop circle, and reduces the driving force from the engines to 1800 N until the speed of the spaceship has reduced to 150 ${\text{m s}}^{-1}$. The work done by the resistive force over this time is 1852.5 kJ.

Find the distance travelled by the spaceship during this time.

A snooker ball of mass 140 g is projected with speed 2.2 ${\text{m s}}^{-1}$ across a rough horizontal surface. It travels 1.5 metres before colliding with the perpendicular wall of the table, and immediately before the collision it is travelling at 1.8 ${\text{m s}}^{-1}$.

Find the coefficient of friction between the ball and the surface.

Another identical ball collides with the wall of the table, travelling at 8 ${\text{m s}}^{-1}$ immediately before the collision. The wall is cushioned and exerts a constant resistive force of 1300 N on the ball, bringing it to rest.

Find the minimum thickness of the wall needed for the ball to stop within the cushion without breaking through the other side, giving your answer in centimetres correct to 3 significant figures.

A lawnmower of mass 9 kg is pushed 8 metres up a hill, along the line of greatest slope, by a force of 80 N acting at an angle of 60° to the slope, as shown in the diagram. The frictional force acting on the lawnmower is 15 N and opposes the motion, parallel to the slope.

Find

(i) the work done against gravity,

(ii) the work done against friction,

(iii) the work done by the 80 N force pushing the lawnmower.

The lawnmower was initially at rest at the bottom of the hill.

Find the speed of the lawnmower at the instant it has been pushed 8 metres up the hill.

Two children in a sledge are pulled up a smooth hill inclined at 10° to the horizontal by a light inextensible rope inclined at 30° to the direction of motion. The tension in the rope has magnitude 120 N, and the sledge moves 15 metres along the line of greatest slope from the point $A$ to the point $B$.

(i) Draw a diagram showing the motion of the sledge between $A$ and $B$, labelling all the forces acting on the sledge.

(ii) Find the total work done by the tension in the rope as the sledge moves from $A$ to $B$.

Once the sledge reaches $B$ it is released from rest and accelerates down the hill. At the point where the sledge has 440 J of kinetic energy, it is moving with speed 4 ${\text{m s}}^{-1}$.

Find the total mass of the sledge and the two children.

Find the speed of the sledge at the instant it reaches $A$ again.

State two assumptions you have made in your answer to part (c).

A package of mass 600 grams is placed on a straight metal slide inclined at 17° to the horizontal. It begins to slide down along the line of greatest slope, gaining speed as it goes. After 25 metres the slide becomes horizontal, and the package travels a further 14 metres along the horizontal section before coming to rest.

Frictional forces act parallel to the motion of the package, and all other non-gravitational resistances are negligible.

Find the coefficient of friction between the package and the metal slide.

Find the greatest speed of the package during the motion.

A load of mass 250 kg is lifted vertically by a cable passing over a pulley. The cable is modelled as light and inextensible, and the pulley as smooth and fixed.

The load starts from rest and is lifted with constant acceleration until it is moving with speed 1.2 ${\text{m s}}^{-1}$. During this time its gravitational potential energy increases by 10 kJ.

Use the work-energy principle to find the work done by the tension in the cable during this time, and hence find the magnitude of the tension.

A diver of mass 50 kg propels herself vertically downwards from a diving board with an initial speed of $v{\text{ m s}}^{-1}$. She moves downwards for 20 metres with constant acceleration before entering the water with a speed of $3v{\text{ m s}}^{-1}$.

(i) Ignoring the effect of air resistance, use energy principles to find the value of $v$.

(ii) Write down the acceleration of the diver as she moves towards the water.

Given that the actual acceleration of the diver is 9.6 ${\text{m s}}^{-2}$ downwards, find the magnitude of the resistance force acting on the diver.

A car is moving up a straight hill, along the line of greatest slope, inclined at $\alpha$° to the horizontal where $\text{sin}\alpha =\frac{3}{5}$. The non-gravitational resistance acting on the car parallel to its motion has magnitude $0.115R$ N, where $R$ is the magnitude of the normal contact force exerted on the car by the ground.

The car travels at constant speed for 1 km, and the work done by the engine during this time is 5250 kJ.

Find the combined mass of the car and its driver.

The car is travelling at 12 ${\text{m s}}^{-1}$ when the driver sees a small dog run into the road ahead and applies the brakes. The total work done against the resistance and the braking force is 21400 J. The braking force acts parallel to the motion of the car and cannot be assumed to be constant.

Find the distance the car travels before coming to rest.

In an experiment a student drops a water balloon vertically from a height above the ground. The balloon falls from rest, and a piece of equipment attached to it beeps first when the balloon's speed reaches $x{\text{ m s}}^{-1}$, and again when its speed is double this. You may assume the equipment beeps for the second time before the balloon hits the ground.

Use energy principles to find the distance between the two beeps in terms of $x$ and $g$.

Given that the second beep sounded exactly 30 metres below the point at which the balloon was dropped, find

(i) the value of $x$,

(ii) the distance the balloon had travelled before the first beep sounded.

State two assumptions you have made in your answers to parts (a) and (b).

A skydiver is falling freely under gravity. At 4000 metres above sea level she is falling at a speed of $v{\text{ m s}}^{-1}$. After falling vertically for a further 100 metres, this speed has doubled.

Modelling the skydiver and her equipment as a particle of mass 80 kg, and ignoring the effect of air resistance, use energy principles to find the value of $v$.

Assuming instead that there is a resistive force, and that the work done against this force during the 100 m fall is $4v$ kJ, use energy principles to find the value of $v$.

Jason needs to move a cement mixer of mass 120 kg into the back of his van, which is at a height of 60 cm above the ground. He has a wooden ramp 1.2 metres long and a metal ramp 2.8 metres long, and he knows from experience that the coefficient of friction between the metal ramp and the mixer is exactly half that between the wooden ramp and the mixer.

You may assume that the mixer begins at rest at the bottom of the ramp and finishes at rest at the top.

Show that less energy is needed to push the cement mixer up the wooden ramp than up the metal ramp.

Given that the work done by the force Jason applies to the cement mixer in pushing it into the van using the wooden ramp is 1850 J, find the coefficient of friction between the cement mixer and the wooden ramp.

Hence find the difference between the amounts of energy needed to push the cement mixer up the two different ramps.

Two loads $A$ and $B$, of masses $m$ kg and $3m$ kg respectively, are attached to the ends of a light inextensible string passing over a fixed smooth pulley. $A$ is at the bottom of a rough plane inclined at $\alpha$° to the horizontal, where $\text{sin}\alpha =0.6$, and $B$ hangs freely at a height of 1.3 m above the level of the bottom of the plane, as shown in the diagram.

The system is released from rest, and in the subsequent motion $A$ moves up the plane along the line of greatest slope.

Given that the net loss in gravitational potential energy of the system when $B$ hits the floor is 12.48 J, find the value of $m$.

The speed of the loads at the point when $B$ hits the ground is exactly 2 ${\text{m s}}^{-1}$ faster than their speed at the instant when $B$ is 1 m above the ground.

Find

(i) the speed with which $B$ hits the ground,

(ii) the magnitude of the frictional force acting on load $A$.

A child of mass 40 kg slides down an uneven slide of varying gradients, of length $l$ metres.

On the first run a constant frictional force of 30 N slows the child down. On the second run the child uses a special mat which makes the effect of the frictional force negligible.

On both runs the child starts from rest at the top. The speed at the bottom is $v_1{\text{ m s}}^{-1}$ without the mat and $v_2{\text{ m s}}^{-1}$ with the mat.

Show that $v_2^{ 2}-v_1^{ 2}=kl$, where $k$ is a constant to be found.

Given that the vertical height of the slide is 25 m, and that the value of $v_2$ is 25% more than the value of $v_1$, find the value of $l$.