In the figure shown a cart moves on a smooth horizontal surface due to an external constant force of magnitude `F`. The initial mass of the cart is `M_(0)` and velocity is zero. Sand falls on to the cart with negligible velocity at constant rate `mu kg//s` and sticks to the cart. The velocity of the cart at time `t` is `:`

Options

A. `(Ft)/(M_(0)+mut)`
B. `(Ft)/(M_(0))e^(mu t)`
C. `(Ft)/(M_(0))`
D. `(Ft)/(M_(0)+mut)e^(mu1)`

Answer

A

Explanation

Fromula `F=m(dv)/(dt)+(V-u)(dm)/(dt)`
Here, `u=`velocity of sand `=O`
`m=M_(o)+mu t =` mass at time `t`
and `(dm)/(dt)=mu`
`:. F=(M_(o)=mu t )(dm)/(dt)+v mu`
`(F-muv)dt=(M_(o)+mut)dv`
`underset(0)overset(t)int(dt)/(M_(o)+mut)=underset(0)overset(v)int(dv)/(F-muv)`
`(1)/(mu)[log (M_(o)+mut)]_(o)^(t)=(1)/(mu)[log(F-muv)]_(o)^(v)`
`"log" ((M_(o)+mut))/(M_(o))= "log" ((F)/(F-mu v))`
`F-muv=(M_(o)F)/(M_(o)=mut)rArrv=(Ft)/(M_(o)+mut)`