A rocket moves vertically upwards in a uniform gravitational field and emits exhaust gas downwards with time-dependent speed relative to the rocket. Derive the rocket equation
where and are respectively the rocket's mass and upward vertical speed at time . Suppose now that and . What is the condition for the rocket to lift off at ? Assuming that this condition is satisfied, find .
State the dimensions of all the quantities involved in your expression for , and verify that the expression is dimensionally consistent.
[ You may assume that all speeds are small compared with the speed of light and neglect any relativistic effects.]