Work, energy and power

A van of mass 900kg is moving along a straight horizontal road.

When the speed of the van is v ms¹, the resistance to the motion of the van is modelled as a force of magnitude 50 v N.

The engine of the van is working at a constant rate of 12 kW.

At the instant when the speed of the van is V ms¹, the acceleration of the van is 0 2. ms²

Using the model,

find the value of V.

Later on, the van is moving up a straight road that is inclined at an angle α to the horizontal, where sin α When the speed of the van is v ms¹, the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v N.

When the van has constant speed U ms⁻¹, the engine of the van is working at a constant rate of 15kW.

Using the model,

find the value of U.

A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle θ to the horizontal, where sin θ The points A and B are on a line of greatest slope of the ramp with AB 18 m and B above A, as shown in Figure 1.

A package of mass 0.5 kg is projected up the ramp from A with speed 8 m s¹ and comes to instantaneous rest at B.

The work done against friction as the package moves from A to B is W joules.

The package is modelled as a particle and air resistance is ignored.

Use the work-energy principle to show that W = 9.7

The coefficient of friction between the package and the ramp is μ

Find the value of μ

A smooth plane is inclined at an angle θ to horizontal ground, where sin θ The points A, B and C lie on a line of greatest slope of the plane, with B between A and C, with A above B and AC = 3a, as shown in Figure 3.

A light elastic string of natural length 2a has one end attached to the fixed point A. The other end of the string is attached to a parcel P of mass m. The modulus of elasticity of the string is mg The parcel rests in equilibrium on the plane at the point B.

The parcel is modelled as a particle.

The parcel is now held at C and released from rest.

Find the elastic potential energy lost by the string when P moves from C to B.

Use the principle of conservation of mechanical energy to find the speed of P at the instant it passes B.

Figure 5 represents the plan view of part of a smooth horizontal floor, where RS and ST are smooth fixed vertical walls. The vector RS→ is in the direction of the vector i. The vector ST→ is in the direction of the vector 3i 4j.

A small ball B of mass 0 25. kg is projected across the floor towards RS.

Immediately before the impact with RS, the velocity of B is v ms¹ Immediately after the impact with RS, the velocity of B is 3i 2j ms₁ The coefficient of restitution between B and RS is The ball is modelled as a particle.

Show that the kinetic energy lost by B in the impact with RS is 4 J.

Immediately after the impact with ST, the velocity of B is w ms¹ Given that the coefficient of restitution between B and ST is

find w in terms of i and j.

A particle P of mass 0.4 kg is moving in a straight line with simple harmonic motion. The maximum speed of P is 5 m s^–1 and the maximum magnitude of acceleration of P is 12.5 m s^–2 The time taken for one complete oscillation is T seconds.

Find the time within each complete oscillation for which P is more than 0.5 m from the centre of the oscillation.

Find the kinetic energy of P at the instant when P is 0.2 m from the centre of the oscillation.

In this question use g = 10 m s^–2

A particle P of mass 0.4 kg is attached to one end of a light inextensible string of length 1.5 m. The other end of the string is attached to a fixed point O. The particle is at rest vertically below O with the string taut.

The particle is then projected horizontally with speed 6 m s^–1

When the string has turned through an angle θ, the string is still taut and the speed of P is vm s^–1, as shown in Figure 8.

[In this question, i and j are horizontal perpendicular unit vectors.]

A particle A has mass 3 kg and a particle B has mass 2 kg.

The particles are moving on a smooth horizontal plane when they collide directly.

Immediately before the collision, the velocity of A is (3i – j) m s^–1 and the velocity of B is (–6i + 2j) m s^–1 Immediately after the collision the velocity of A is 2i m s

Find the total kinetic energy of the two particles before the collision.

Find, in terms of i and j, the impulse exerted on A by B in the collision.

Find, in terms of i and j, the velocity of B immediately after the collision.

A rough plane is inclined to the horizontal at an angle θ, where tanθ =

A particle P of mass m is at rest at a point on the plane.

The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance d. The coefficient of friction between P and the plane is

Show that the magnitude of the frictional force acting on P as it moves up the plane

Air resistance is assumed to be negligible.

Using the work-energy principle,

find d in terms of a.

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of 72 km h^–1

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of PkW.

Using the model,

find the value of P.

The car now travels in a straight line up a road which is inclined to the horizontal at an

• In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude 20v newtons, where vm s^–1 is the speed of the car

At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at 10 m s^–1, the acceleration of the car is am s^–2

Using the refined model,

find the value of a.

Later on, when the engine of the car is again working at a constant rate of 30 kW, the car is moving up the road at a constant speed Um s^–1

Using the refined model,

find the value of U.

A particle A of mass 2m is moving in a straight line with speed 3u on a smooth horizontal plane. Particle A collides directly with a particle B of mass m which is at rest on the plane.

The coefficient of restitution between A and B is e, where e 0

Show that the speed of B immediately after the collision is 2u(1 + e).

After the collision, B hits a smooth fixed vertical wall which is perpendicular to the direction of motion of B.

Show that there will be a second collision between A and B.

The coefficient of restitution between B and the wall is

Find, in simplified form, in terms of m, u and e,

the magnitude of the impulse received by B in its collision with the wall,

the loss in kinetic energy of B due to its collision with the wall.

A light elastic string has natural length 2a and modulus of elasticity 2mg. One end of the string is attached to a fixed point A on a horizontal ceiling. The other end is attached to a particle P of mass m.

The particle P hangs in equilibrium at the point E, where AE = 3a. The particle P is then projected vertically downwards from E with speed

Air resistance is assumed to be negligible.

Find the elastic energy stored in the string, when P first comes to instantaneous rest. Give your answer in the form kmga, where k is a constant to be found.

A smooth solid hemisphere has radius r and the centre of its plane face is O. The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6. A small stone is at the point A, the highest point on the surface of the hemisphere. The stone is projected horizontally from A with speed U. The stone is still in contact with the hemisphere at the point B, where OB makes an angle θ with the upward vertical. The speed of the stone at the instant it reaches B is v.

The stone is modelled as a particle P and air resistance is modelled as being negligible.

Use the model to find v² in terms of U, r, g and θ

When P leaves the surface of the hemisphere, the speed of P is W.

After leaving the surface of the hemisphere, P moves freely under gravity until it hits the ground.

Find the speed of P as it hits the ground, giving your answer in terms of r and g.

At the instant when P hits the ground it is travelling at α to the horizontal.

Find the value of α.

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed U m s⁻¹. The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW.

Find the value of U.

The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N. The resistance to the motion of the trailer is a constant force of magnitude 300 N.

The engine of the car is working at a constant rate of 16 kW.

The tow rope is modelled as being light and inextensible.

Using the model,

A light elastic string has natural length 2 a and modulus of elasticity 4 mg. One end of the elastic string is attached to a fixed point O. A particle P of mass m is attached to the other end of the elastic string. The particle P hangs freely in equilibrium at the point E, which is vertically below O

Find the length OE.

Particle P is now pulled vertically downwards to the point A, where OA = 4a, and released from rest. The resistance to the motion of P is a constant force of

Find, in terms of a and g, the speed of P after it has moved a distance a.

Particle P is now held at O Particle P is released from rest and reaches its maximum speed at the point B. The resistance to the motion of P is again a constant force of magnitude mg.

Find the distance OB.

A particle P of mass m is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle α, where 0 < 45 At the instant immediately before the impact, the speed of P is u.

At the instant immediately after the impact, P is moving horizontally with speed v.

Show that the magnitude of the impulse exerted on the plane by P is mu sec

The coefficient of restitution between P and the plane is e, where e > 0

Show that v² u²(sin² e² cos²)

Show that the kinetic energy lost by P in the impact is

Hence find, in terms of m, u and e only, the kinetic energy lost by P in the impact.

A smooth hemisphere of radius a is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is O.

A particle P of mass M is disturbed from rest at the highest point of the hemisphere.

When P is still on the surface of the hemisphere and the radius from O to P is at an angle θ to the vertical,

• the speed of P is v

• the normal reaction between the hemisphere and the particle is R, as shown in Figure 2.

Find, in terms of a and g, the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

end of the string is attached to a fixed point O. The package hangs at rest vertically below O with the string taut and is then projected horizontally with speed u, as shown in Figure 5.

When OP has turned through an angle θ and the string is still taut, the tension in the string is T

The package is modelled as a particle and the string as being light and inextensible.

Given that P moves in a complete vertical circle with centre O

find, in terms of a and g, the minimum possible value of u

find, in terms of g, the magnitude of the acceleration of P at the instant when OP is horizontal.

Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.

Throughout this question, use g = 10 m s^–2

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N.

A particle P of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point A. Particle P hangs freely in equilibrium with P vertically below A

The particle is then pulled vertically down to a point B and released from rest.

Show that, while the string is taut, P moves with simple harmonic motion with

The maximum kinetic energy of P during the subsequent motion is 2.5 J.

Show thatAB = 2 m

The particle returns to B for the first time T seconds after it was released from rest at B

Find the value of T

A van of mass 900 kg is moving along a straight horizontal road.

At the instant when the speed of the van is vm s⁻¹, the resistance to the motion of the van is modelled as a force of magnitude (500 + 7 v) N.

When the engine of the van is working at a constant rate of 18 kW, the van is moving along the road at a constant speed Vm s⁻¹

Find the value of V.

Later on, the van is moving up a straight road that is inclined to the horizontal at an

At the instant when the speed of the van is vm s⁻¹, the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude (500 + 7 v) N.

The engine of the van is again working at a constant rate of 18 kW.

Find the acceleration of the van at the instant when v = 15

Two particles, A and B, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle A has mass 5m and particle B has mass 3m.

The coefficient of restitution between A and B is e, where e 0

Immediately after the collision the speed of A is v and the speed of B is 2v.

Given that A and B are moving in the same direction after the collision,

find the set of possible values of e.

Given also that the kinetic energy of A immediately after the collision is 16% of the kinetic energy of A immediately before the collision,

find

the value of e,

the magnitude of the impulse received by A in the collision, giving your answer in terms of m and v.

Figure 1 represents the plan view of part of a horizontal floor, where AB and BC represent fixed vertical walls, with AB perpendicular to BC.

A small ball is projected along the floor towards the wall AB. Immediately before hitting the wall AB the ball is moving with speed vm s⁻¹ at an angle θ to AB.

The ball hits the wall AB and then hits the wall BC. The coefficient of restitution between the ball and the wall AB is

The coefficient of restitution between the ball and the wall BC is e.

The floor and the walls are modelled as being smooth.

The ball is modelled as a particle.

The ball loses half of its kinetic energy in the impact with the wall AB.

Find the exact value of cosθ.

The ball loses half of its remaining kinetic energy in the impact with the wall BC.

Find the exact value of e.

A light elastic spring has natural length 3l and modulus of elasticity 3mg.

One end of the spring is attached to a fixed point X on a rough inclined plane.

The other end of the spring is attached to a package P of mass m. The plane is inclined to the horizontal at an angle α where tanα =

The package is initially held at the point Y on the plane, where XY = l. The point Y is higher than X and XY is a line of greatest slope of the plane, as shown in Figure 2.

The package is released from rest at Y and moves up the plane. The coefficient of friction between P and the plane is

By modelling P as a particle,

show that the acceleration of P at the instant when P is released from rest is g

find, in terms of g and l, the speed of P at the instant when the spring first reaches its natural length of 3 l.

A light inextensible string of length a has one end attached to a fixed point O. The other end of the string is attached to a small stone of mass m. The stone is held with the string taut and horizontal. The stone is then projected vertically upwards with speed U.

The stone is modelled as a particle and air resistance is modelled as being negligible.

Assuming that the string does not break, use the model to

find the least value of U so that the stone will move in complete vertical circles.

The string will break if the tension in it is equal to

find the total angle that the string has turned through, from when the stone is projected vertically upwards, to when the string breaks,

find the magnitude of the acceleration of the stone at the instant just before the string breaks.

A light elastic string, of natural length l and modulus of elasticity 2mg, has one end attached to a fixed point A and the other end attached to a particle P of mass m. The particle P hangs in equilibrium at the point O.

The particle P is pulled down vertically from O to the point B, where OB = l, and released from rest.

Air resistance is modelled as being negligible.

Using the model,

prove that P begins to move with simple harmonic motion about O with

The particle P first comes to instantaneous rest at the point C.

Using the model,

find the length BC in terms of l,

find, in terms of l and g, the exact time it takes P to move directly from B to C.

A truck of mass 1200 kg is moving along a straight horizontal road.

At the instant when the speed of the truck is vm s^–1, the resistance to the motion of the truck is modelled as a force of magnitude (900 + 9 v) N.

The engine of the truck is working at a constant rate of 25 kW.

Find the deceleration of the truck at the instant when v = 25

Later on, the truck is moving up a straight road that is inclined at an angle θ to the

At the instant when the speed of the truck is vm s^–1, the resistance to the motion of the truck from non‑gravitational forces is modelled as a force of magnitude (900 + 9 v) N.

When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of Vm s^–1.

Find the value of V.

A smooth uniform sphere P has mass 0.3 kg. Another smooth uniform sphere Q, with the same radius as P, has mass 0.2 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of P is (4i + 2j) m s^–1 and the velocity of Q is (–3i + j) m s^–1.

At the instant of collision, the line joining the centres of the spheres is parallel to i.

The kinetic energy of Q immediately after the collision is half the kinetic energy of Q immediately before the collision.

Find

the velocity of P immediately after the collision,

the velocity of Q immediately after the collision,

the coefficient of restitution between P and Q,

carefully justifying your answers.

Find the size of the angle through which the direction of motion of P is deflected by the collision.

A light elastic string with natural length l and modulus of elasticity kmg has one end attached to a fixed point A on a rough inclined plane. The other end of the string is attached to a package of mass m. The plane is inclined at an angle θ to the horizontal, where tanθ =

The coefficient of friction between the package and the plane is

By modelling the package as a particle,

find the acceleration of the package at the instant it starts to move back down the plane from the point B.

Figure 2 represents the plan view of part of a horizontal floor, where AB and CD represent fixed vertical walls, with AB parallel to CD.

A small ball is projected along the floor towards wall AB. Immediately before hitting wall AB, the ball is moving with speed vm s^–1 at an angle α to AB, where 0 < α < The ball hits wall AB and then hits wall CD. After the impact with wall CD, the ball is moving at angle α to CD. The coefficient of restitution between the ball and wall AB is The coefficient of restitution between the ball and wall CD is also The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

A particle P of mass m is attached to one end of a light inextensible string of length l. The other end of the string is attached to a fixed point O. The particle is held with the string taut and OP horizontal. The particle is then projected vertically downwards with speed u, where u²= gl. When OP has turned through an angle α and the string is still taut, the speed of P is v, as shown in Figure 5. At this instant the tension in the string is T.

Find, in terms of g and l, the speed of P at the instant when the string goes slack.

Find, in terms of l, the greatest vertical height reached by P above the level of O.

A light elastic spring has natural length l and modulus of elasticity 4mg. A particle P of mass m is attached to one end of the spring. The other end of the spring is attached to a fixed point A. The point B is vertically below A with AB= . The particle P is released from rest at B.

Find, in terms of m, l and g, the maximum kinetic energy of P during the motion.

Find the time within each complete oscillation for which the length of the spring is less than l.