Conservation & dissipation of energy

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.

A particle of mass m moves in a circle of radius r. The number of revolutions completed per second is f.

What is the kinetic energy of the particle?

In 2021 the world land speed record was 1230 km h⁻¹. This was the average speed achieved by a jet-powered car in two runs. Each run was measured over a distance of 1.61 km.

The average speed for one of these runs was 343 m s⁻¹.

Calculate, in s, the time taken for the car to complete the other run.

Engineers are designing a new jet-powered car to break this record.

Figure 1 shows the variation of speed with distance for the car, as predicted by the engineers.

The car reaches its maximum acceleration when it is 5600 m from the start. At this point the mass of the car is 6.50 × 10³ kg.

Determine the kinetic energy of the car at its maximum acceleration.

At any point on the graph in Figure 1, the acceleration is given by:

acceleration = speed × gradient of line

When the car is at its maximum acceleration, the power input to the jet engines is 640 MW.

Calculate the percentage of the input power used to accelerate the car at its maximum acceleration.

Scientists recommend that the average deceleration of the driver of the car should be less than 3 g.

Deduce whether the average deceleration is less than 3 g.

A car travels at 100 km h⁻¹ on a motorway.

What is an estimate of its kinetic energy?

A photon has energy of 1 × 10¹⁸ eV. An object of mass 0.03 kg has kinetic energy equal to the energy of the photon.

What is the speed of the object?

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?

An electric motor lifts a load of weight W through a vertical height h in time t. The potential difference across the motor is V and the current in it is I.

What is the efficiency of the motor?

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.

An electric motor lifts a load of weight W through a vertical height h in time t. The potential difference across the motor is V and the current through it is I.

What is the efficiency of the motor?

An object of mass m moves in a circle of radius r. It completes n revolutions every second.

What is the kinetic energy of the object?

Horizontal escape lanes made of loose gravel have been constructed at the side of some roads on steep hills so that vehicles can stop safely when their brakes fail.

Figure 1 shows an engineer’s prediction of how the speed of an unpowered vehicle of mass 1.8 × 10⁴ kg will vary with time as the vehicle comes to rest in an escape lane.

Determine the force decelerating the vehicle 2.0 s after entering the escape lane.

Deduce whether a lane of length 85 m is long enough to stop the vehicle, assuming that the engineer’s graph is correct.

Discuss the energy transfers that take place when a vehicle is decelerated in an escape lane.

An alternative to an escape lane containing gravel is an escape lane that consists of a ramp. An escape ramp is a straight road with a concrete surface that has a constant upward gradient.

One escape ramp makes an angle of 25° to the horizontal and is 85 m long.

Deduce whether this escape ramp is sufficient to stop the vehicle.

Assume that any frictional forces and air resistance that decelerate the vehicle are negligible.

Discuss whether an escape lane containing gravel or an escape ramp would provide the safer experience for the driver of the vehicle as it comes to rest.