Question

A partical of mass `m` is driven by a machine that deleveres a constant power `k` watts. If the partical starts from rest the force on the partical at time `t` is
A. `sqrt((mk)/(2))t^(-1//2)`
B. `sqrt(mk)t^(-1//2)`
C. `sqrt(2mk)t^(-1//2)`
D. `1/2sqrt(mk)t^(-1//2)`

Answer 1

Correct Answer - a
(a) As the machine delivers a constant power so `F.v="constant"=k("watts")`
`impliesm(dv)/(dt).v=k`
`implies intvdv=k/(m)intdt`
`implies (v^(2))/(2)=(k)/(m)timpliesv=sqrt((2k)/(m))t`
Now, force on the particle is given by
`F=m(dv)/(dt)=m(d)/(dt)((2kt)/(m))^(1/2)`
`=sqrt(2km).(1/(2)t^(-1/2))=sqrt((mk)/(2)).t^(-1/2)`