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

Explanation

Let the particle is moving on a curvilinear path. When it has travelled a distances, the force Facting on it and its speed v are shown in the adjoining figure.

Instantaneous power delivered by the engine: `P = vecF . vecv = (vec(F_r) + vec(F_N)) . vecv = F_T v = ma_T V`
Tangential acceleration of the vehicle: `a_T = v (dV)/(ds) = k sqrt(s) . d/(ds) (ksqrt(s)) = (k^2)/2`
From above equations, we have `P = F_T v = ma_T v = m. (k^2)/(2). k sqrt(s) = (mk^2)/(2) sqrt(s)`