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

Explanation

We have `(dm)/(dt)=-mu` or, `dm=-mudt`
Integrating `underset(m_0)overset(m)intdm=-muunderset(0)overset(t)intdt`, or, `m=m_0-mut`
As `vecu=0` so, from the equation of variable mass system:
`(m_0-mut)(dvecv)/(dt)=vecF` or, `(dvecv)/(dt)=vecw=vecF//(m_0-mut)`
or, `underset(0)overset(vecv)intdvecv=vecFunderset(0)overset(t)int(dt)/((m_0-mut))`
Hence `vecv=(vecF)/(mu)1n((m_0)/(m_0-mut))`