Inverse-square gravitational force

2. A particle \(P\) of mass \(m\) is at a distance \(x\) above the surface of the Earth. The Earth exerts a gravitational force on \(P\). This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of \(P\) from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }\) A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance \(R\) above the surface of the Earth, the particle has speed \(U\). Air resistance is modelled as being negligible.
2. Find, in terms of \(U , g\) and \(R\), the speed of the particle when it strikes the surface of the Earth.
   

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1. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.

The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)

• the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
• the speed of \(S\) is \(v\)
  1. Show that

$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)

Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.

2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.

1. Show that \(K = m g R ^ { 2 }\) (2) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.

5. A particle \(P\) of mass 0.4 kg moves on the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { x ^ { 2 } }\) newtons, where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 2\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(x = 5\) the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(k\). The particle first comes to instantaneous rest at the point \(A\).
2. Find the value of \(x\) at \(A\).

1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
   At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.

At time \(t\) seconds, where \(t \geqslant 0\)

• \(\quad P\) is a distance \(x\) from \(O\)
• \(P\) is moving with speed \(v\)
• \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)

Air resistance is modelled as being negligible.

1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.

2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.

1. Show that \(K = m g R ^ { 2 }\) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.

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5. Above the Earth's surface, the magnitude of the gravitational force on a particle due to the Earth is inversely proportional to the square of the distance of the particle from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and the acceleration due to gravity at the Earth's surface is \(g\). A particle \(P\) of mass \(m\) is at a height \(x\) above the surface of the Earth.

1. Show that the magnitude of the gravitational force acting on \(P\) is $$\frac { m g R ^ { 2 } } { ( R + x ) ^ { 2 } }$$ A rocket is fired vertically upwards from the surface of the Earth. When the rocket is at height \(2 R\) above the surface of the Earth its speed is \(\sqrt { } \left( \frac { g R } { 2 } \right)\). You may assume that air resistance can be ignored and that the engine of the rocket is switched off before the rocket reaches height \(R\). Modelling the rocket as a particle,
2. find the speed of the rocket when it was at height \(R\) above the surface of the Earth.

2. A particle \(P\) of mass \(m\) is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When \(P\) is at a distance \(x\) from the centre of the Earth, the gravitational force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that \(k = m g R ^ { 2 }\). When \(P\) is at a height \(\frac { R } { 4 }\) above the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 2 } }\) Given that air resistance can be ignored,
2. find, in terms of \(R\), the greatest distance from the centre of the Earth reached by \(P\).

3. A particle \(P\) of mass \(m\) moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\). When \(P\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). When \(P\) is at a distance \(2 R\) from the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 3 } }\). Assuming that air resistance can be ignored, find the distance of \(P\) from the surface of the Earth when the speed of \(P\) is \(2 \sqrt { \frac { g R } { 3 } }\).

3. A particle \(P\) of mass 0.5 kg moves away from the origin \(O\) along the positive \(x\)-axis under the action of a force directed towards \(O\) of magnitude \(\frac { 2 } { x ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres. When \(x = 1\), the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when its speed has been reduced to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
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5. When a particle of mass \(M\) is at a distance of \(x\) metres from the centre of the moon, the gravitational force, \(F\) N, acting on it and directed towards the centre of the moon is given by $$F = \frac { \left( 4.90 \times 10 ^ { 12 } \right) M } { x ^ { 2 } }$$ A rocket is projected vertically into space from a point on the surface of the moon with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the radius of the moon is \(\left( 1.74 \times 10 ^ { 6 } \right) \mathrm { m }\),

1. show that the speed of the rocket, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is \(x\) metres from the centre of the moon is given by $$v ^ { 2 } = u ^ { 2 } + \frac { a } { x } - b$$ where \(a\) and \(b\) are constants which should be found correct to 3 significant figures.
2. Find, correct to 2 significant figures, the minimum value of \(u\) needed for the rocket to escape the moon's gravitational attraction.