Gravitational Potential & Energy (OCR A Level Physics)

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.

i) Indicate with a letter X on the orbit where the asteroid would be moving at maximum speed.

[1]

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

[2]

The gravitational potential at a distance r from the centre of the asteroid Ryugu is V_g. The graph of V_g against for the asteroid is shown below.

i) Define gravitational potential.

[1]

ii) 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.

[1]

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

M = ...................................... kg [2]

In October 2018, the probe Mobile Asteroid Surface Scout (MASCOT) was released from rest from the Hayabusa2 spacecraft from a distance of 600 m from the centre of the asteroid.

Assume that the spacecraft was stationary relative to the asteroid when MASCOT was dropped.

Use information from (b) to calculate the speed of the impact v when MASCOT landed on the surface of the asteroid.

v = ................................................ m s^–1 [3]

This question is about a space probe which is in orbit around the Sun.

Define the gravitational potential of an object at a point in a gravitational field.

[1]

The space probe has mass 810 kg. The orbital radius of the space probe is 1.5 × 10¹¹ m. The orbital period of the space probe around the Sun is 3.16 × 10⁷ s. The mass of the Sun is 2.0 × 10³⁰ kg.

i) Show that the magnitude of the gravitational potential energy of the space probe is about 7 × 10¹¹ J.

[2]

ii) Show that the kinetic energy of the space probe is half the value of your answer to (b)(i).

[3]

iii) Calculate the total energy of the space probe.

total energy = ....................................................... J [1]

The power source for the instrumentation on board the space probe is plutonium-238, which provides 470 W initially.

Plutonium-238 decays by α-particle emission with a half-life of 88 years. The kinetic energy of each α-particle is 8.8 × 10^–13 J.

i) Calculate the number N of plutonium-238 nuclei needed to provide the power of 470 W.

N = ......................................................... [3]

ii) Calculate the power P still available from the plutonium-238 source 100 years later.

P = ..................................................... W [3]