Kinetic, elastic and gravitational potential energy

An asteroid of mass 10 kg is in a circular orbit around the Sun at a distance of 3.5 AU from the centre of the Sun.

1 AU = 1.5 × 10¹¹ m

Use Kepler’s third law to calculate the period of the orbit of the asteroid in years.

An object orbiting the Sun at a distance of 1 AU has a period of 1 year.

Calculate the kinetic energy of the asteroid.

Use the graph to calculate the change in gravitational potential energy of the satellite from the moment of collision until it crosses the Earth’s orbit.

Eventually, the asteroid falls into a new orbit around the Sun. This orbit is elliptical.

State the other feature of this orbit predicted by Kepler’s first law.

This question is about comparing two methods for storing energy.

A proposed method for storing energy uses trucks powered by electric motors.

The trucks carry sand up a mountain, storing surplus energy as gravitational potential energy.

When the energy needs to be released, they carry the sand back down the mountain, converting the gravitational potential energy into chemical energy stored in a battery.

Explain how an electrical generator in the truck could use electromagnetic induction to convert gravitational potential energy into chemical energy stored in a battery.

A second proposed method for storing energy uses a tank of wax.

The wax is heated using an electrical heater, storing surplus energy as internal energy of the wax. The wax increases in temperature and melts as it is heated.

When the energy needs to be released, the wax is allowed to cool, converting the internal energy of the wax into thermal energy.

Explain, in terms of the particles in the wax, how energy is stored when it is heated.

Gravitational waves are disturbances in space-time that are produced by violent astrophysical events.

Define strain.

The maximum strain shown on the graph is approximately 10^–21. (ii)

Calculate the force that would produce the same strain in a copper wire of diameter 2 mm.

Young modulus of copper = 130 GPa

By considering the answer to (ii), suggest why it is very difficult to detect the strain caused by gravitational waves.

In this part of the question you will use Newtonian physics to estimate the power output of the black hole collision.

The diagram shows two identical black holes, each of mass M, orbiting each other with period T.

Prior to colliding, the black holes move at speed V in identical circular orbits of diameter D about the central point C.

Write down an expression for the gravitational force, F, acting between the black holes in terms of M, D and the universal gravitational constant, G.

The total orbital energy, E, of the black holes is the sum of their kinetic energy and gravitational potential energy. Show that E is given by the expression:

E = k (²G²M⁵)^1/3 (1/T^2/3)

where k is a numerical constant.

By considering times B and D, use your answer to (iii) to estimate the power radiated as gravitational waves in this time interval.

Describe how to find the centre of mass of a 2-dimensional shape, including any equipment required.

Fig. 19 shows a bridge.

The bridge can be raised by an electric motor to allow tall ships to pass underneath.

The moving section of the bridge is 14 m long and has a weight of 120 kN.

The centre of mass of the structure is 6.3 m from the pivot.

Calculate the average power required to raise the bridge to a vertical position in 90 s.

Suggest why the actual electric motor used to lift the bridge has a maximum power output several times larger than the value calculated in (b)(i).

An athlete is running at a speed of about 5 m s^–1.

What is a reasonable estimate for the kinetic energy of this athlete?

A 12 J

B 100 J

C 900 J

D 800 000 J

Your answer

An archer fires an arrow towards a target as shown below.

The diagram is not drawn to scale.

The centre of the target is at the same height as the initial position of the arrow. The target is a distance of 90 m from the arrow. The arrow has an initial velocity of 68 m s^–1 and is fired at an angle of 11 to the horizontal.

Air resistance has negligible effect on the motion of the arrow.

Describe how the kinetic energy of the arrow changes during its journey from when it is fired until it reaches its maximum height.

Show that the time taken for the arrow to reach its maximum height is about 1.3 s.

The arrow misses the target.

Calculate the horizontal distance, measured along the base line, by which the arrow misses the target.

Explain what is meant by an inelastic collision.

The target is mounted on wheels. The target has a much larger mass than the mass of the arrow.

Using ideas of momentum, explain the velocity of the target immediately after the arrow sticks into the target.

In June 2018, the spacecraft Hayabusa2 arrived at an asteroid called Ryugu.

The asteroid orbits the Sun in an elliptical orbit as shown below.

The diagram is not drawn to scale.

Indicate with a letter X on the orbit where the asteroid would be moving at maximum

speed.

Use Kepler’s second law to explain your answer to (a)(i).

Define gravitational potential.

Show that the magnitude of the gradient of the graph is equal to GM, where M is the mass of the asteroid and G is the gravitational constant.

Use the gradient of the graph to show that the mass M of the asteroid is about 4.6 × 10¹¹ kg.

The Planck constant h is an important fundamental constant in quantum physics.

Determine the S.I. base units for h.

A researcher is investigating the de Broglie wavelength of charged particles.

The charged particles are accelerated through a potential difference V. The de Broglie wavelength of these particles is then determined by the researcher.

Each particle has mass m and charge q.

A beam of -particles is incident on a thin gold foil. Most -particles pass straight through the foil. A few are deflected by gold nuclei.

The diagram shows the path of one -particle which passes close to a gold nucleus N in the foil. The -particle is deflected through an angle of 60° as it travels from A to B.

P marks its position of closest approach to the gold nucleus.

Another -particle in the beam is deflected by the same gold nucleus N through an angle of 30°.

Sketch its path onto the diagram above.

The initial kinetic energy of each -particle is 5.0 MeV.

Show that the magnitude of the initial momentum of each -particle is about 10^–19 kg m s^–1 Take the mass of the -particle to be 6.6 × 10^–27 kg.

Draw two vectors in the spaces below to represent the initial momentum and the final momentum of the -particle.

initial momentum at A

final momentum at B

Draw a vector in the space below to represent the momentum of the nucleus N when the -particle reaches B.

Explain how you determined this momentum.

A pendulum bob is oscillating in a vacuum. The maximum height of the bob from the ground is 1.3 m and its minimum height is 1.1 m.

What is the maximum speed of the pendulum bob?

2.0 m s^–1 A

3.9 m s^–1 B

5.1 m s^–1 C

26 m s^–1 D

Your answer

Explain the shape of the curve in Fig. 3.3.

The torch is inverted 80 times.

Calculate the total energy stored in the capacitor.

Fig. 4.1 shows an arrangement used by a student to determine the acceleration of free fall.

A trolley is attached to a variable mass m by a string which passes over a pulley.

The mass m is released from rest and falls through a fixed height of 0.600 m accelerating the trolley of mass 0.800 kg. When the mass m hits the floor, the trolley then continues to move at a constant velocity v.

This constant velocity v is determined by measuring the time t for the card of length 0.200 m to pass fully through a light gate connected to a timer.

Frictional forces on the trolley and the falling mass m are negligible.

Show that the relationship between v and m is

where g is the acceleration of free fall.

The student records the information from the experiment in a table. The column headings and just the last row for m = 0.600 kg from this table are shown below.

It is suspected that the card on the trolley did not pass at right angles through the light beam.

Discuss, without doing any calculations, the effect this may have on the experimental value for the acceleration of free fall g.

Calculate the work done by the 400 N force exerted by the mechanic.

Calculate the useful work done on the car and hence the percentage efficiency of the jack.