Momentum

A person kicks a stationary football horizontally.

During the kick, the person gives an impulse of 6.55 N s to the ball during a contact time of 12.0 ms. The mass of the football is 410 g. Assume that friction forces are negligible.

Calculate the kinetic energy transferred to the ball during the kick.

Figure 8 shows how the force F acting on the ball varies with time t.

Determine the scale for the F / N axis and add your scale to Figure 8.

A player jumps up to head a football when it is 2.0 m above the ground. The initial velocity of the ball is 18 m s⁻¹ at an angle of 48° to the horizontal.

Figure 9 shows the path of the ball above horizontal ground.

Determine the time that the ball takes to reach the ground from the instant it leaves the player’s head.

Assume that the frictional forces acting on the ball are negligible.

The deuterium–tritium (D–T) reaction is a nuclear reaction between two isotopes of hydrogen. The D–T reaction is

2 3 4

The energy from this reaction is transferred to the kinetic energy of the helium nucleus and the kinetic energy of the neutron. Assume that the kinetic energies of the hydrogen nuclei are zero just before the reaction occurs.

Show that the kinetic energy of the neutron represents approximately 80% of the total energy transferred.

The combined kinetic energy of the helium nucleus and the neutron is 2.82 × 10^–12 J.

Calculate the initial speed of the neutron.

A cable system is to be used to transfer supplies across a river. A model of the proposed system is built in order to test its performance.

The model consists of a cable attached to two vertical posts A and B, as shown in Figure 3. A pulley P of negligible mass is attached to the cable.

In this question the length of the cable does not change and the weight of the cable can be ignored.

An object O is attached to P. In one test, O and P are at rest in the position shown in Figure 3. The weight of O is 350 N.

Figure 4 is a force diagram drawn to scale. It represents the magnitudes and directions of the tensions T₁ and T₂ in the cable when O is at rest in the position shown in Figure 3. At this position, resistive forces are zero.

Complete the force diagram. Go on to determine, using your diagram, the magnitudes of T₁ and T₂.

Figure 4

In a second test, pulley P with O attached is released from A. P and O move along the cable to B.

The change in height of the centre of mass of O between A and B is 4.5 m. The distance travelled along the cable is 18 m.

The speed of O when it reaches B is 6.5 m s⁻¹.

Calculate the average resistive force on O and P as they move from A to B.

O contains a fragile item packed in suitable material.

Explain how the material can prevent damage to the fragile item when O stops suddenly at B.

An average force of 42 kN acts on the air passing through a jet engine. This force causes the speed of the air to increase by 540 m s⁻¹.

What mass of air passes through the engine in one minute?

An object is moving in a straight line. A graph is plotted to show the variation of the momentum of the object with time.

Which quantities can be calculated from the gradient of the graph and the area under

A variable force F acts on an object of mass 2.0 kg. The object is at rest at time t = 0

The graph shows the variation of F with t.

What is the speed of the object when t = 1.0 s ?

A tennis ball has a mass of 58 g. The ball is dropped from rest from a height of 1.8 m above the ground and falls vertically. The ball rebounds vertically to a height of 1.1 m. The effect of air resistance is negligible.

What is the change in momentum of the ball during its collision with the ground?

A coin is projected horizontally from the top of a desk. The diagram shows the coin at one point in its path. The air resistance is negligible.

The arrows E, F and G represent different directions.

Which row gives the direction of the acceleration and the direction of the momentum of the coin at this point?

A golf ball has a mass of 46 g and is initially stationary. The diagram shows the variation with time of the force acting on the golf ball as it is hit with a golf club.

What is an estimate of the kinetic energy of the golf ball immediately after it is hit?

A pellet with velocity 200 m s⁻¹ and mass 5.0 g is fired vertically upwards into a stationary block of mass 95.0 g. The pellet remains in the block. The impact causes the block to move vertically upwards.

What is the maximum vertical displacement of the block?

Safety barriers are used on UK motorways to prevent vehicles crossing from one carriageway to the other carriageway. The barriers also absorb some of the kinetic energy of a vehicle and deflect vehicles along the barrier.

The standard test of a safety barrier uses a vehicle that contains dummies. The total mass of the vehicle and its contents is 1.5 × 10³ kg and its initial speed is 110 km h^–1

Show that the initial kinetic energy of the test vehicle is 700 kJ.

The test vehicle hits a steel safety barrier at an angle of 20°, as shown in Figure 7.

Calculate the component of the momentum of the test vehicle in a direction along the line of the safety barrier. Give an appropriate unit for your answer.

Immediately after the collision, the test vehicle moves along the safety barrier with no change in its momentum in this direction.

Show that the kinetic energy lost in the collision is about 80 kJ.

The steel safety barrier deforms during the collision. For the barrier to pass the test, the test vehicle should not move more than 1.5 m towards the other carriageway.

The barrier can apply an average force of 60 kN at right angles to the carriageway.

Deduce whether the safety barrier will pass the test.

A different safety barrier uses a solid concrete wall which does not deform. The same standard test is carried out on a concrete wall.

Discuss which type of barrier would cause less damage to the dummies in the test.

A body of constant mass falls freely due to gravity.

The rate of change of momentum of the body is equal to its

Figure 10 shows two railway trucks A and B travelling towards each other on the same railway line which is straight and horizontal.

The trucks are involved in an inelastic collision. They join when they collide and then move together.

The trucks move a distance of 15 m before coming to rest.

Truck A has a total mass of 16 000 kg and truck B has a total mass of 12 000 kg

Just before the collision, truck A was moving at a speed of 2.8 m s^–1 and truck B was

State the quantity that is not conserved in an inelastic collision.

Show that the speed of the joined trucks immediately after the collision is about

Calculate the impulse that acts on each truck during the collision. Give an appropriate unit for your answer.

Explain, without doing a calculation, how the motion of the trucks immediately after the collision would be different for a collision that is perfectly elastic.

What is a correct unit for the area under a force − time graph?

Figure 8

The model consists of two identical trolleys of mass M on a ramp which is at 35° to the horizontal. The trolleys are connected by a wire that passes around a pulley of negligible mass at the top of the ramp.

Two concrete blocks each of mass m are loaded onto trolley A at the top of the ramp. The trolley is released and accelerates to the bottom of the ramp where it is stopped by a flexible buffer. The blocks are unloaded from trolley A and two blocks are loaded onto trolley B that is now at the top of the ramp. The trolleys are released and the process is repeated.

Figure 9 shows the side view of trolley A when it is moving down the ramp.

The tension in the wire when the trolleys are moving is T.

Draw and label arrows on Figure 9 to represent the magnitudes and directions of any forces and components of forces that act on trolley A parallel to the ramp as it travels down the ramp.

Assume that no friction acts at the axle of the pulley or at the axles of the trolleys and that air resistance is negligible.

Show that the acceleration a of trolley B along the ramp is given by

Compare the momentum of loaded trolley A as it moves downwards with the momentum of loaded trolley B.

In practice, for safety reasons there is a friction brake in the pulley that provides a resistive force to reduce the acceleration to 25% of the maximum possible acceleration.

The distance travelled for each journey down the ramp is 9.0 m.

The following data apply to the arrangement.

Mass of a trolley M = 95 kg Mass of a concrete block m = 30 kg

Calculate the time taken for a loaded trolley to travel down the ramp.

It takes 12 s to remove the blocks from the lower trolley and reload the upper trolley.

Calculate the number of blocks that can be transferred to the lower level in 30 minutes.

What is the thrust of the rocket?

Following the collision, the van moves with a velocity of 6.20 m s⁻¹ and the car recoils in the opposite direction with a velocity of 1.60 m s⁻¹.

What is the initial speed of the car?