A constant force F is applied on a trolley of initial mass `m_0` kept over a smooth surface. Sand is poured gently over the trolley at a constant rate of `(mu)`kg//s`. Afer time t, find

(a) mass of the trolley (with sand)
(b) net force on the trolley
(c) velocity of trolley

Explanation

(a) Mass of the trolley after time t is
`m=mass of trolley +(mass of sand poured per second)(time)`
`=m_0+mut`
(b) Let v is the velocity of trolley at time t. Sand is poured gently. So, velocity of sand is zero or, relative veloctiy of sand is v in the opposite direction of velocity of trolley. Mass of trolley is increasing. So, thrust force on the trolley is in the direction of relative velocity or in the opposite direction of motion of trolley.

Net force on trolley `=F-F_1`
`=F-v_r(dm)/(dt)=F-(v)(mu)`
or `F_(n et)=F-muv`
In the above expression fo `F_(net')` velocity `v` is a function of time which has been asked in the next part.
(c) `F_(n et)=F-vmu`
`ma=F-muv`
`implies(m_0+mut)((dv)/(dt))=F-muv`
`rArr int_0^v(dv)/(F-muv)=int_0^t(dt)/(m_0+mut)`
Solving this equation we get
`v=(Ft)/(m_0_mut)`