Inverse power force - gravitational/escape velocity context

3. Above the earth's surface, the magnitude of the force on a particle due to the earth's gravity is inversely proportional to the square of the distance of the particle from the centre of the earth. Assuming that the earth is a sphere of radius \(R\), and taking \(g\) as the acceleration due to gravity at the surface of the earth,

1. prove that the magnitude of the gravitational force on a particle of mass \(m\) when it is a distance \(x ( x \geq R )\) from the centre of the earth is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the earth with initial speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g R\). Ignoring air resistance,
2. find, in terms of \(g\) and \(R\), the speed of the particle when it is at a height \(2 R\) above the surface of the earth.

3. A rocket is fired vertically upwards with speed \(U\) from a point on the Earth's surface. The rocket is modelled as a particle \(P\) of constant mass \(m\), and the Earth as a fixed sphere of radius \(R\). At a distance \(x\) from the centre of the Earth, the speed of \(P\) is \(v\). The only force acting on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { c m } { x ^ { 2 } }\), where \(c\) is a constant.

1. Show that \(v ^ { 2 } = U ^ { 2 } + 2 c \left( \frac { 1 } { x } - \frac { 1 } { R } \right)\). The kinetic energy of \(P\) at \(x = 2 R\) is half of its kinetic energy at \(x = R\).
2. Find \(c\) in terms of \(U\) and \(R\).
   (3)

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

1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the Earth with initial speed \(3 U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).
2. Find \(U\) in terms of \(g\) and \(R\).

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 a magnitude which is inversely proportional to \(x^2\). 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 acting on \(P\) is \(\frac{mgR^2}{x^2}\) [2]

The particle was fired with initial speed \(U\) and the greatest height above the surface of the Earth reached by \(P\) is \(\frac{R}{20}\). Given that air resistance can be ignored,

1. find \(U\) in terms of \(g\) and \(R\). [7]

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

1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac{mgR^2}{x^2}\). [3]

A particle is fired vertically upwards from the surface of the Earth with initial speed \(3U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).

1. Find \(U\) in terms of \(g\) and \(R\). [7]

A projectile \(P\) is fired vertically upwards from a point on the earth's surface. When \(P\) is at a distance \(x\) from the centre of the earth its speed is \(v\). Its acceleration is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant. The earth may be assumed to be a sphere of radius \(R\).

1. Show that the motion of \(P\) may be modelled by the differential equation $$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]

The initial speed of \(P\) is \(U\), where \(U^2 < 2gR\). The greatest distance of \(P\) from the centre of the earth is \(X\).

1. Find \(X\) in terms of \(U\), \(R\) and \(g\). [6]

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

1. Show that \(k = mgR^2\). [2]

Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,

1. find the speed of \(S\) as it crashes into the surface of the earth. [7]

A particle of mass \(m\) kg, at a distance \(x\) m from the centre of the Earth, experiences a force of magnitude \(\frac{km}{x^2}\) N towards the centre of the Earth, where \(k\) is a constant. Given that the radius of the Earth is \(6.37 \times 10^6\) m, and that a 3 kg mass experiences a force of 30 N at the surface of the Earth,

1. calculate the value of \(k\), stating the units of your answer. [3 marks]

The 3 kg mass falls from rest at a distance \(x = 12.74 \times 10^6\) m from the centre of the Earth. Ignoring air resistance,

1. show that it reaches the surface of the Earth with speed \(7.98 \times 10^3\) ms\(^{-1}\). [7 marks]

In a simplified model, the particle is assumed to fall with a constant acceleration 10 ms\(^{-2}\). According to this model it attains the same speed as in (b), \(7.98 \times 10^3\) ms\(^{-1}\), at a distance \((12.74 - d) \times 10^6\) m from the centre of the Earth.

1. Find the value of \(d\). [3 marks]