1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) in terms of \(t\).
3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).