5 A particle P of mass \(m \mathrm {~kg}\) is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a rough horizontal surface. During the motion of P , a constant frictional force of magnitude \(F \mathrm {~N}\) acts on P . When the velocity of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a force of magnitude \(k v \mathrm {~N}\) due to air resistance, where \(k\) is a constant.

1. Determine the dimensions of \(k\). At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is
   \(\mathrm { T } = \frac { \mathrm { mu } } { \mathrm { F } } - \frac { \mathrm { kmu } ^ { 2 } } { 2 \mathrm {~F} ^ { 2 } } + \frac { 1 } { 3 } \mathrm { k } ^ { 2 } \mathrm {~m} ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { F } ^ { \gamma }\).
2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).