A flat car of mass `m_(0)` starts moving to the right due to a constant horizontal force `F.S` and spills on the flat car from a stationary hopper. The rate of loading is constant and equal to `mukg//s`. Find the time dependence of the velocity and the acceleration of the flat car in the process of loading. the friction is negligibly small.

Explanation

`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_0 mut) 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))`