Question

A cart loaded with sand moves along a horizontal floor due to a constant force $F$ coinciding in direction with the cart's velocity vector. In the process, the sand spills through a hole in the bottom with constant rate $\mu kg/s$. Find the acceleration and the velocity of the cart at the moment $t$, if the initial moment $t=0$ the cart with loaded sand had mass $m_0$ and its velocity was equal to zero. Friction is to be neglected.

Answer 1

In this problem, the sand spills through a hole at the bottom of the cart. Hence, the relative velocity of the sand ${\nu }_r$ will be zero, because it will acquire the same velocity as that of the cart at the moment of separation.
${\nu }_{rel}=0$
Apply $F+\nu rel\frac{dm}{dt}=m\frac{d\nu }{dt}$
$⟹F+0\times \left(\frac{dm}{dt}\right)=(m_0-\mu t)\frac{d\nu }{dt}$ (i)
$⟹(m_0-\mu t)\frac{d\nu }{dt}=F$
or ${\int }_0^{\nu }d\nu ={\int }_0^t\frac{Fdt}{m_0-\mu t}$
$⟹\nu =\frac{F}{-\mu }\left[In\right(m_0-\mu t){]}_0^t$
or $/nu=\frac{F}{\mu }In\left(\frac{m_0}{m_0-\mu t}\right)$
From Eq. (i) acceleration of the cart.
$a=\frac{d\nu }{dt}=\frac{F}{m}$ or $a=\frac{F}{m_0-\mu t}$