A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).
Show that, while the spaceship is ejecting fuel,
$$\frac { \mathrm { d } v } { \mathrm {~d} m } = - \frac { c } { m } .$$
The initial mass of the spaceship is \(m _ { 0 }\) and at time \(t\) the mass of the spaceship is given by \(m = m _ { 0 } ( 1 - k t )\), where \(k\) is a positive constant.
Find the acceleration of the spaceship at time \(t\).