A flatcar of mass `m_0` starts moving to the right due to a constant horizontal force F(figure). Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to `mukg//s`. Find the time dependence of the velocity and the acceleration of the flatcar in the process of loading. The friction is negligibly small.

Explanation

Let the car be moving in a reference frame to which the hopper is fixed and at any instant of time, let its mass be m and velocity `vecv`.
Then from the general equation, for variable mass system.
`m(dvecv)/(dt)=vecF+vecu(dm)/(dt)`
We write the equation, for our system as,
`m(dvecv)/(dt)=vecF-vecv(dm)/(dt)` as, `vecu=-vecv` (1)
So `(d)/(dt)(vecmv)=vecF`
and `vecv=(vecFt)/(m)` on integration.
But `m=m_0+mut`
so, `vecv=(vecFt)/(m_0(1+(mut)/(m_0))`
Thus the sought acceleration, `vecw=(dvecv)/(dt)=(vecF)/(m_0(1+(mut)/(m_0))^2`