Show that the acceleration

\frac{dv}{dt}

of a particle moving in a straight line can be written, in terms of its velocity

v

and its displacement

x

from a point of the line, in the form

v \frac{dv}{dx}

At a distance $x$ km from the centre of the Earth the gravitational acceleration in km s $^{- 2}$ is given by the formula $c / x^{2}$ where $c = 4 \times 10^{5}$ . If a space craft $10^{4}$ km from the centre of the Earth is moving directly away from it at a speed of 10 km s $^{- 1}$ at what distance will it be moving with half that speed?