5 In this question you may assume that if \(x\) and \(y\) are any physical quantities then \(\left[ \frac { \mathrm { dy } } { \mathrm { dx } } \right] = \left[ \frac { \mathrm { y } } { \mathrm { x } } \right]\). A machine drives a piston of mass \(m\) into a vertical cylinder. The equation below is suggested to model the power developed by the machine, \(P\), while it is not doing any other external work. $$\mathrm { P } = \mathrm { k } _ { 1 } \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dt } } + \mathrm { k } _ { 2 } \mathrm { mgv } + \mathrm { k } _ { 3 } \mathrm { E }$$ in which

• \(v\) is the velocity of the piston at a given time,
• \(g\) is the acceleration due to gravity,
• \(E\) is the rate at which heat energy is lost to the surroundings,
• \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are dimensionless constants.

Determine whether the equation is dimensionally consistent. Show all the steps in your argument.