Question

Show how the alternative definition of power, found in your book, can be derived by substituting the definitions of work and speed into the standard
definition of power, P = W /delta t

Answer 1

Let us consider body moves a distance S due to the force F.

Hence the work by the body W = FS

If the force is not along the direction of displacement,then the work by a body for travelling a distance S will be -

                                       $W=[ Fcos\theta]*S$  where    $Fcos\theta$ is the component of the force along the direction of displacement.

                                  $Hence\ W= FScos\theta$

                                                        $= F.S$

As per the question the power P is given as -

                                                  $P=\frac{W}{\delta t}$

                                                         $=\frac{F.S}{\delta t}$

                                                         $= F.\frac{S}{\delta t}$

                                                         $= \ F.V$

Hence alternative definition of power P = F.V

Answer 2

Power formula is P= W/ Delta t, or we know that W= saclar product of D and F, D=length of distance traveled, F= the applied force, it is W = //D// //F// cos(D, F), when we substitute, we get P=[ //D// //F// cos (D, F)]/ deltat, or //D// = V x delta t, P= [V x delta t x //F// cos(D, F)] / delta t, and then P= V//F//cos(D, F), or V and D have the same direction, so cos(D, F)= cos(V, F), finally  P= V. F (or scalar product of V and F)