A flat car of mass m_(0) starts moving to the right due to a constant horizontal force. Sand spills on the flat car from a stationary hopper . The velocity of loading is constant and equal to mu kg//s . Find the time dependance of the velocity and the acceleration of the flat car in the process of loading . The friciton is negligible small .

A flat car of mass m_(0) starts moving to the right due to a constant

1.	A flat car of mass m_(0) starts moving to the right due to a constant horizontal force. Sand spills on the flat car from a stationary hopper . The velocity of loading is constant and equal to mu kg//s . Find the time dependance of the velocity and the acceleration of the flat car in the process of loading . The friciton is negligible small .
Answer» Solution :`m(dv)/(dt)+V(dm)/(dt)=F` Mass of the car at any instant `=m_(0)+mu t` `:. (m_(0)+MUT)(dv)/(dt)+vmu=F` `:. (dv)/(F-muv)=(dt)/(m_(0)+mut)` Integrating , we have log`(F-muv)=-log(m_(0)+mut)+C` when`t=0,v=0` `:.Log (F-muv)=-log(m_(0)+mut)+logF+log m_(0)` `:. (F-muv)/F=m_(0)/(m_(0)+mut)` `(muv)/F=(mut)/(m_(0)+mut)["SINCE" if a/B=C/d,(b-a)/b=(d-C)/d] "or" v=(FT)/(m_(0)+mut)` `:.` Acceleration `=(dv)/(dt)=d/(dt)((Ft)/(m_(0)+mut))=((m_(0)+mut)F-Ftmu)/((m_(0)+mut))^(2)` `=(Fm_(0)+muFt-muFt)/((m_(0)+mut))^(2)=(Fm_(0))/(m_(0)^(2)(1-:(mut)/m_(0))^(2))=F/(m_(0)(1-:(mut)/m_(0)))`