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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

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Nirina7

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)

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


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