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A body is thrown with a velocity v0 at a...

A body is thrown with a velocity `v_0` at an angle `theta_0` with horizontal. Find the (a) instantaneous power delivered by gravity after a time t measured from the instant of projection and (b) average power delivered by gravity during the time t.

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(i) Power : Power delivered by gravity after a time t when the particle has velocity v is,
`P_(g) = m vec(g).vec(v)" "...(i)`
Kinematics : Substituting `vec(g) = - g vec(j) and vec(v) = v_(0) cos theta_(0) hat(i) + (v_(0) sin theta_(0) - "gt")`j in eq. (i),
We have `P_(g) = - mg (v_(0) sin theta_(0) - "gt")`
(ii) The average power is `P_(av) = (int_(0)^(t) Pdt)/(t)`
Substituting `P = - mg (v_(0) sin theta_(0) -"gt")`, we have `P_(av) = - (mg)/(t) int_(0)^(t) (v_(0) sin theta_(0) -"gt")dt`
This gives `P_(av) = - mg(v_(0) sin theta_(0) - ("gt")/(2))`
Note : If t = T (time of light), then `P_(av)` = zero
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