Work & Energy (CIE A Level Maths: Mechanics)

A company uses a cable to move a crate of mass 50 kg up a 20 metre ramp inclined at 40° to the horizontal by applying a force of 400 N as shown in the diagram below.  The frictional force acting on the crates is  where is the coefficient of friction between the crate and the ramp.

Given that for a crate to move the full way up the ramp, the work done against friction is equal to 1500 J, find

the magnitude of the normal reaction, ,

the coefficient of friction, , giving your answer correct to 2 decimal places. 

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

The cable exerts a force of 400 N in the direction of motion of the crate, calculate the net work done by all the forces acting on the crate including the force due to gravity.

A cyclist is travelling at constant speed of 10 m s^-1   on horizontal ground when they see traffic up ahead and apply their brakes.  

Modelling the cyclist and their bicycle as a single particle, find the kinetic energy of the cyclist in terms of their total mass,  kg, before applying their brakes.

The cyclist slows down over 16 metres with a constant deceleration and then continues moving at a constant speed of  m s^-1 .

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

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

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

The bus then accelerates with a driving force of against a constant resisting force of 206 .  It reaches 15 m s^-1 again after travelling 300 m.  

Calculate the magnitude of .

State an assumption you have made about the driving force, .

A single force acts on a particle of mass kg which makes it accelerate from an initial speed of  m s^-1 to a speed of  m s^-1. During this time, the particle travels 20 metres over a flat horizontal surface.  

Use the work-energy principle to show that the magnitude of the work done by the force acting on the particle can be given as  

Given that the gain in kinetic energy is 250 J and that the force acting on the particle can be assumed to be constant, 

find the acceleration in terms of .

A person pulls their narrow boat 5 metres along a flat, horizontal canal against a total resistance to motion of 750 N.  They are standing alongside the canal and are holding the rope at an angle of  to the direction of motion.

Draw a diagram showing the forces acting on the narrow boat.

Find the work done against the resistance to motion.

The tension in the rope when is .  Given that the boat is moving with constant velocity in the direction of motion, find the value of .

To move around a tree stump in the canal the person changes the angle of the rope to 30°.  Find the increase in tension on the rope needed to ensure the boat continues moving at the same speed.

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 m s^-1 to 18 m s^-1.  

Calculate:

State whether any external forces, other than gravity, were doing work on the car and give a reason for your answer.

A skateboarder moves off from rest from the top of a ramp and passes through the point  with a speed of 2 m s^-1.  The point  is 4 m vertically below .

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

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  if the skateboarder and equipment can be modelled as a particle of mass   60 kg.

A toboggan team consisting of two children and their sled are moving down a 400 m long uneven course of varying gradients.  They start from rest at the top of the course and do no work as they move down the course.  The frictional force on the sled is 8 N and all other resistance forces can be ignored.  The vertical height of the hill is 35 m.  Find the speed that the team is moving at when they reach the end of the course.  You may consider the two children and their sled as a single particle of mass 100kg.

A box of mass 4 kg slides down the line of greatest slope on a ramp inclined at  to the horizontal with constant acceleration 2 m s^-2.  Two markers on the ramp are set 7 metres apart and the box slides past the second marker exactly 2 seconds after sliding past the first.  

Show that the box passes the first marker with speed 1.5 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, 

the value of the angle, ,

A package with a mass of 600 grams is placed on a straight metal slide inclined at 17° to the horizontal.  It starts to slide down the hill along the line of greatest slope and gains speed as it moves down.  After sliding for 25 metres the slide turns horizontal and the package continues sliding for a further 14 metres until it eventually comes to a stop.  There are frictional forces acting parallel to the motion of the package and all other non-gravitational resistance forces 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 using a cable on a pulley system.  The cable is modelled as light and inextensible, and the pulley is modelled as smooth and fixed. The load starts from rest and is lifted with constant acceleration until it is moving with speed 1.2 m s^-1.  During this time, the 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.

Superpig of mass 150 kg is flying horizontally with a constant velocity of  m s^-1.  She propels herself forwards using a driving force of 2000 N from her own specially designed light jet pack which includes wings upon which a lift force is acting vertically upwards. 

Draw a diagram showing the forces acting on Superpig, include the magnitudes of the lift force and the resistive force from air resistance in your diagram.

Find the work done by each of the four forces acting on Superpig over a ten second period given that =200.

Superpig spots a possible villainous pig in the distance and reduces the driving force from her engines to 1800 N until her speed has reduced to 150 m s^-1.  The work done by the resistive force over this time is 1852.5 kJ. 

Find the distance travelled by Superpig during this time.

A snooker ball of mass 140 g is projected with a speed of 2.2 m s^-1 across a rough horizontal surface.  It travels 1.5 metres before colliding with the perpendicular wall of the table. Immediately before the collision the ball is travelling at 1.8 m s^-1 .

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

Another identical ball collides with the wall of the table, immediately before this collision the speed of the ball is 8 m s^-1. The wall of the table is cushioned and exerts a constant resistive force of 1300 N on the ball to bring it to rest.

Find the minimum thickness of the wall of the table to ensure the ball stops within the cushion and does not break through the other side.  Give your answer in cm to 3 significant figures.

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

The lawnmower is pushed 8 metres up the hill. 

Calculate:

Given that the lawnmower was initially at rest at the bottom of the hill, find the speed of the lawnmower in the instant that it has been pushed 8 metres up the hill.

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

Ignoring the effect of air resistance, use energy principles to find the value of .

Given that the actual acceleration of the diver is 9.6605 ms^-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 steepest slope, inclined at to the horizontal where .  The non-gravitational resistance force acting on the car parallel to the motion of the car has a magnitude of 0.115 , where  is the magnitude of the normal reaction force exerted on the car by the ground.  The car travels at a constant speed for 1 km and the work done by the engine of the car during this time is 5250 kJ. 

Find the combined mass of the car and its driver.

The car is travelling with speed 12 m s^-1 when the driver of the car sees a small dog run into the road just ahead and applies the brakes.  The total work done against the resistance force and the braking force is 21400 J.  The braking force acts parallel to the motion of the car and it can not be assumed to be constant.

Find the distance the car travels before coming to rest.

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 that is 1.2 metres long and a metal ramp that is 2.8 metres long and he knows from experience that the coefficient of friction between the metal ramp and the cement mixer is exactly half of that of the wooden ramp.  You may assume in your working that the mixer begins at rest at the bottom of the ramp and finishes its motion at rest at the very top of the ramp.

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

Given that the work done by the force applied by Jason to the cement mixer in order to push it into his 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 children in a sledge are being 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 as it is being pulled has magnitude 120 N and the sledge moves forward 15 metres along the line of greatest slope from the point to the point .

Draw a diagram showing the motion of the sledge between the points and , labelling all forces acting on the sledge.

Calculate the total work done by the tension in the rope as the sledge is moved up the slope from to .

Once the sledge reaches point B it is then 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 of 4 m s^-1.  Find the total mass of the sledge and the two children.

Two loads and , of masses  kg and 3 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley.  is at the bottom of a rough plane inclined at to the horizontal, where  and is hanging freely at a height of 1.3 m above the level of the bottom of the plane as shown in the diagram below.  The system is released from rest in the subsequent motion moves up the plane along the line of greatest slope.

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

The speed of the loads at the point when  hits the ground is exactly 2 m s^-1 faster than the speed of the loads at the instant when  is 1 m above the ground.  Find

the speed with which  hits the ground,

the magnitude of the frictional force acting on particle . 

In an experiment a student drops a water balloon vertically from a height above ground.  She allows the balloon to fall from rest and a piece of equipment attached to the balloon beeps first at the point when the balloon’s speed reaches  m s^-1 and again when it is double this speed. You may assume that the equipment beeps for a second time before the water balloon hits the ground.

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

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

the value of ,

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

A child of mass 40 kg slides down an uneven slide, length  metres, of varying gradients.  The child was annoyed by a constant frictional force of 30 N slowing them down so they returned with a special mat which causes the effect of the frictional force to be negligible.  Both of the times the child used the slide they moved off from rest at the top and their speeds at the bottom of the slide were  m s^-1 for the slide without the special mat and  m s^-1 for the slide with the special mat.

Show that   where  is constant to be found.

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

A skydiver is falling freely under gravity.  At 4000 metres above sea-level she is falling at a speed of  m s^-1 and after falling vertically for 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 .

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