5 A particle of mass \(m\) moves in a straight line with constant acceleration \(a\). Its initial and final velocities are \(u\) and \(v\) respectively and its final displacement from its starting position is \(s\). In order to model the motion of the particle it is suggested that the velocity is given by the equation
\(\mathrm { v } ^ { 2 } = \mathrm { pu } ^ { \alpha } + \mathrm { qa } ^ { \beta } \mathrm { s } ^ { \gamma }\)
where \(p\) and \(q\) are dimensionless constants.

1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent.
2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\).
3. By considering the case where \(s = 0\), determine the value of \(p\).
4. By multiplying both sides of the equation by \(\frac { 1 } { 2 } m\), and using the numerical values of \(\alpha , \beta\) and \(\gamma\), determine the value of \(q\).