Two particles, of masses \(m\) and \(2 m\), are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude \(k v ^ { 2 }\), where \(v\) is the speed of each particle after it has moved a distance \(x\) from rest and \(k\) is a positive constant.
Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }\)
Find \(v ^ { 2 }\) in terms of \(x\).
Deduce that the tension in the string, \(T\), satisfies
$$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$