A vehicle of mass m starts moving such that its speed v varies with distance a according to the equation `v = ksqrt(s)`, where k is a positive constant. Deduce a relation to express the instantaneous power delivered by its engine.

Explanation

Let the particle is moving on a curvilinearpath. When it has travelled a distance s, the force F acting on it and its speed v are shown in the adjoining figure.

Instantaneous power delivered by the engine : `P = vecF. vecv = (vecF_(r) + vecF_(N)). vecv = F_(tau)v = ma_(tau)v`
Tangential acceleration of the vehicle : `a_(tau) = v (dv)/(ds) = ksqrt(s)., (d)/(ds) (ksqrt(s)) = (k^(2))/(2)`
From above equations , we have `P = F_(tau)v = ma_(tau)m. (k^(2))/(2). ksqrt(s) = (mk^(3))/(2) sqrt(s)`