3 A particle \(P\) of mass \(m\) moves on the \(x\)-axis under the action of a force \(F\) directed along the axis. When the displacement of \(P\) from the origin is \(x\) its velocity is \(v\).
- By using the fact that the dimensions of the derivative \(\frac { d v } { d x }\) are the same as those of \(\frac { v } { x }\), verify that the equation \(\mathrm { F } = \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dx } }\) is dimensionally consistent.
It is given that \(\mathrm { v } = \mathrm { km } ^ { - \frac { 1 } { 2 } } \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } }\) where \(a\) and \(k\) are constants.
- Explain why \([ a ]\) must be the same as \([ x ]\).
- Deduce the dimensions of \(k\).
- Find an expression for \(F\) in terms of \(x\) and \(k\).