A particle of mass m is driven by a machine that delivers a constant power k watts. If the partilcle starts from rest the force on the particle at time t is

Options

A. `sqrt(2 mk) t^(-1//2)`
B. `1/2 sqrt(mk) t^(-1//2)`
C. `sqrt((mk)/(2)) t^(-1//2)`
D. `sqrt(mk) t^(-1//2)`

Answer

C

Explanation

Constant power acting on the particle of mass m is k watt.
or `P = k or (dW)/(dt) = k, dW = kdt`
Integrating both sides, `int_0^W dW = int_0^t k dt`
`implies W = kt`
Using work energy theorem, `W = 1/2 mv^2 - 1/2 m(0)^2`
`kt = 1/2 mv^2 " or " v = sqrt((2kt)/(m))`
Acceleration of the particle, `a = (dv)/(dt) = 1/2 sqrt((2k)/(m) = 1/(sqrtT)) = sqrt((k)/(2mt))`
Force on the particle , `F = ma = sqrt((mk)/(2t)) = sqrt((mk)/(2)) t^(-1//2)`.