A flat car of mass m, 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 friction is negligibly small.

A flat car of mass m, starts moving to the right due to a constant hor

1.	A flat car of mass m, 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 friction is negligibly small.
Answer» Solution :`m(dv)/(dt)+ v(dm)/(dt) = F ` MASS of the car at any instant `= m_0 + mut ` `:. ( m_0 + mu t) (dv)/(dt) + v mu =F` `:. (dv)/(F- mu v) =(dt)/(m_0 + mu t)` Integrating , we have `log (F- mu v) = - log ( m_0 + mu t) + C` when t=0 , v=0 `:. log F = - log m_0 + C` `:. log (F-muv) = - log ( m_0 + mu t) + log F + log m_0` `:. (F_ mu v)/( F) = (m_0)/( m_0 + mu t)` `(mu v)/(F) = (mu t)/(m_0 + mu t)[" SINCE " if (a)/(b) = (C)/(d) ,(b-a)/(b) =(d-C)/(d)] or v=(FT)/(m_0 + mu t)` `:.` Acceleration `=(dv)/(dt) = (d)/(dt) ( Ft)/( m_0 + mu t) = ((m_0mut) F-F t mu)/((m_0 + mut)^(2))` `=(F m_0 + mu F t - mu Ft)/( (m_0 + mut)^(2)) = (F m_0)/(m_0^(2)(1 + (mut)/(m_0))^(2))=(F)/(m_0 (1 + (mut)/(m_0))`