6 A particle moves in a straight line with constant acceleration \(a\). Its initial velocity is \(u\) and at time \(t\) its velocity is \(v\).
It is assumed that \(v\) depends only on \(u , a\) and \(t\).
- Assuming that this dependency is of the form \(\mathrm { u } ^ { \alpha } \mathrm { a } ^ { \beta } \mathrm { t } ^ { \gamma }\), use dimensional analysis to find \(\alpha\) and \(\gamma\) in terms of \(\beta\).
- By noting that the graph of \(v\) against \(t\) must be a straight line, determine the possible values of \(\beta\).
You may assume that the units of the given quantities are the corresponding SI units.
- By considering \(v\) when \(t = 0\) seconds and when \(t = 1\) second, derive the equation of motion \(\mathrm { v } = \mathrm { u } + \mathrm { at }\), explaining your reasoning.