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 rate `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

In this problem the sand spills through a hole at the bottom of the cart. Hene ,the relative velocity of the sand `v_(r)` will be zero, because it will acquire the same velocity as that of the cart at the moment of separation.

`v_(rel)=0`
Apply `F=v_(rel)(dm)/(dt)=m(dv)/(dt)`
`implies F+0xx((dm)/(dt))=(m_(0)-mut)(dv)/(dt)` .............i
`implies(m_(0)-mut)(dv)/(dt)=F`
or `int_(0)^(v)dv=int_(0)^(t)(Fdt)/(m_(0)-mut)`
`implies v=F/(-mu)[In(m_(0)-mut)]_(0)^(t)`
or `v=F/muIn(m_(0)/(m_(0)-mut))`
Frome eqn i acceleration of the cart
`a=(dv)/(dt)=F/m` or `a= F/(m_(0)-mut)`