Gravitational potential

In February 2021 the spacecraft Perseverance Rover landed on Mars. When the spacecraft was 11.0 km above the surface of Mars, parachutes opened to slow the descent. The parachutes detached from the spacecraft when it was 2.1 km above the surface of Mars.

Calculate the change in gravitational potential energy of the spacecraft during the parachute section of its descent.

mass of spacecraft = 1030 kg mass of Mars = 6.39 10²³kg radius of Mars = 3390 km

Astronomers observing stars at the centre of our galaxy have suggested that many of them are orbiting a supermassive black hole. The mass of this black hole is 9.2 10³⁶kg.

Calculate the orbital period for a star in a circular orbit at a distance of 1.9 10¹⁴m from a black hole of this mass.

The star S0-2 is in a highly elliptical orbit around the position of the black hole.

At its point of closest approach, S0-2 is at a distance of 1.8 10¹³m from the centre of the black hole.

At the most distant point of its orbit, S0-2 is 2.7 10¹⁴m from the black hole.

Show that the change in gravitational potential between the closest and most distant points in this orbit is about 3 10¹³J kg⁻¹.

At its point of closest approach, the star is travelling at a speed of 8.1 10⁶m s⁻¹.

Calculate the speed of S0-2 at the furthest point in its orbit using the change in gravitational potential.

mass of S0-2 = 2.4 10³¹kg

Trigonometric parallax and Hubble’s law are two methods used to determine astronomical distances. Explain whether either of these methods is suitable to determine the distance to S0-2.

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

In 1990, the Hubble Space Telescope (HST) was launched into a low Earth orbit above the Earth’s atmosphere.

HST orbits the Earth in a circular orbit with a speed of 7.59 km s^–1.

mass of Earth = 5.97 × 10²⁴ kg radius of Earth = 6.37 10⁶ m

Show that the height of HST above the surface of the Earth is about 6 × 10⁵ m.

Calculate the increase in the gravitational potential energy as HST is raised, from its initial position at the Earth’s surface, to its orbital height.

mass of HST = 11 600 kg

A student suggests that giving HST more energy than that required in (ii) would result in the satellite orbiting at a greater height and with a greater speed.

Assess the validity of the student’s suggestion.

The transmission of electromagnetic radiation through the atmosphere is shown on the graph.

State one advantage shown by this graph of positioning a telescope above the atmosphere.

High resolution images from HST allow astronomers to make detailed measurements of very distant galaxies. The graph shows how the recessional velocities of distant galaxies depend on their distance from Earth.

Determine an age for the universe.