5 The escape speed of an unpowered object is the minimum speed at which it must be projected to escape the gravitational influence of the Earth if it is projected vertically upwards from the Earth's surface. A formula for the escape speed \(U\) of an unpowered object of mass \(m\) is \(U = \sqrt { \frac { 2 G m } { r } }\) where \(r\) is the radius of the Earth and \(G\) is a constant.
- Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\).
A rocket is a powered object. A rocket is launched with a given launch speed and is then powered by engines which apply a constant force for a period of time after the launch.
A student wishes to apply the formula given above to a rocket launch. They wish to model the minimum launch speed required for a rocket to escape the Earth’s gravitational influence.
They realise that the given formula is for unpowered objects and so they include an extra term in the formula to obtain \(V = \sqrt { \frac { 2 G m } { r } } - \mathrm { kP } ^ { \alpha } \mathrm { W } ^ { \beta } \mathrm { t } ^ { \gamma }\).
In their modified formula, \(G\) and \(r\) are the same as before. The other variables are defined as follows.
- \(V\) is the required minimum launch speed of the rocket
- \(k , \alpha , \beta\) and \(\gamma\) are dimensionless constants
- \(P\) is the power developed by the engines of the rocket
- \(m\) is the initial mass of the rocket
- \(W\) is the initial weight of the rocket
- \(t\) is the total time for which the engines of the rocket operate
- Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
- By considering the value of \(\gamma\) found in part (b) explain the relationship between \(t\) and \(V\).