A body of mass `m` is thrown at an angle `alpha` to the horizontal with the initial velocity `v_0` Find (a) instantaneous power of gravity as a function of time and
(b) mean power developed by gravity in time (i) `t=0` to `T`,
(ii) `t=0` to `(T)/(2)`, (iii) `t=(T)/(2)` to `t=T`, where `T` is the time of flight.

Explanation

(a) Force on body `vecF=-mghatj`
velocity at time `t`,`vecv=v_0cosalphahati+(v_0sinalpha-g t)hatj`
Instantaneuos power delivered by the gravitational force
`P=vecF.vecv=-mg(v_0sinalpha-g t)`
`=mg(g t-v_0sinalpha)`
(b) Average power `overlineP=(W)/(t)`
(i) `t=0` to `t=T`, work done by gravitational force `=0` (displacement in the direction of force `=0`)
(ii) `t=0` to `t=(T)/(2)`, `W=-mgH_(max)`
W: work done by gravitational force
`overlineP=(-mgH_(max))/((T)/(2))=-mg``((v_0^2sin^2alpha)/(2g))/((v_0sinalpha)/(g))`
`=(-mgv_0sinalpha)/(2)`
(iii) `t=(T)/(2)` to `t=T`, `W=mgH_(max)`
`overlineP=(mgH_(max))/((T)/(2))=(mgv_0sinalpha)/(2)`