A ballistic missile, fired vertically from the Earth’s surface with an initial velocity of u just reaches a height h.
Using energy methods, derive an expression giving u² in terms of R (the radius of the Earth), g and h. Assume the gravitational acceleration at a distance r from the centre of the Earth is $\frac{GM}{r^{2}}$ , where GM = gR².

Question

A ballistic missile, fired vertically from the Earth’s surface with an initial velocity of u just reaches a height h.
Using energy methods, derive an expression giving u² in terms of R (the radius of the Earth), g and h. Assume the gravitational acceleration at a distance r from the centre of the Earth is [latex]\frac{GM}{r^{2}}[/latex] , where GM = gR².

Accepted Answer

At launch, the kinetic energy of the missile is [latex]\frac{1}{2}mu^{2}[/latex] and its potential energy is [latex]- \frac{GMm}{R}.[/latex]

At the missile’s highest point, its velocity is zero and since it is a distance (R + h) from the centre of the Earth, the potential energy is [latex]- \frac{GMm}{R + h}.[/latex]
So, using the principle of conservation of energy:

loss in kinetic energy = gain in potential energy

[latex]\frac{1}{2}mu^{2} = GMm \left(- \frac{1}{R + h} + \frac{1}{R}\right)[/latex]

= GMm[latex] \left(\frac{h}{R(R + h)}\right)[/latex]

⇒ u² = [latex]\frac{2gRh}{(R + h)}[/latex] since GM = gR².

This is the result obtained previously on page 409.

Note
Conservative forces
It is shown on the previous page that the work done against the gravitational force depends only on the initial and final position of the body. This means that when the body returns to its starting point, no net work has been done, so the kinetic energy will be the same as before.
That is, the force conserves the total mechanical energy. Forces which have this property are known as conservative. It can be shown that conservation of mechanical energy is equivalent to the fact that the work done is dependent only on the initial and final positions.
Potential energy is associated only with conservative forces. The gravitational force, the electrical force between charged particles, the tension in a spring (obeying Hooke’s law) – these are all examples of conservative forces. Frictional forces and the forces involved in nonelastic collisions are not conservative. They are known as dissipative as they result in a reduction of the total mechanical energy .

Question 18.6: A ballistic missile, fired vertically from the Earth’s surfa...

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