Question

A flatcar of mass $m_0$ starts moving to the right due to a constant horizontal force $\overrightarrow{F}$ (figure shown above). Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to $\mu kg/s$. Find the time dependence of the velocity and the acceleration of the flatcar in the process of loading. The friction is negligibly small.

Answer 1

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 $\overrightarrow{v}$.
Then from the general equation, for variable mass system,
$m\frac{d\overrightarrow{v}}{dt}=\overrightarrow{F}+\overrightarrow{u}\frac{dm}{dt}$
We write the equation, for our system as,
$m\frac{d\overrightarrow{v}}{dt}=\overrightarrow{F}-\overrightarrow{v}\frac{dm}{dt}$ as, $\overrightarrow{u}=-\overrightarrow{v}$ (1)
So, $\frac{d}{dt}\left(\overrightarrow{m}v\right)=\overrightarrow{F}$
and $\overrightarrow{v}=\frac{\overrightarrow{F}t}{m}$ on integration.
But $m=m_0+\mu t$
so, $\overrightarrow{v}=\frac{\overrightarrow{F}t}{m_0(1+\frac{\mu t}{m_0})}$
Thus the sought acceleration, $\overrightarrow{w}=\frac{d\overrightarrow{v}}{dt}=\frac{\overrightarrow{F}}{m_0(1+\frac{\mu t}{m_0})}$