# A Cart losing Mass

Tags: cart, losing, mass
 P: 14 Hi I have this problem involving a cart which is losing sand It says: A cart with initial mass $$M$$ and a load of sand $$\frac{1}{2}M$$ loses sand at the rate $$k$$ kg/s. The cart is pulled horizontally by a force $$F$$. Find the differential equation for the rate of change of the carts velocity in terms of $$k,M$$ and $$F$$ while there is sand in the cart. So i said that at $$t = 0$$ the momentum$$= \frac{3}{2}Mv$$. Therefore $$\displaystyle{dp = \left(\frac{3}{2}Mv\right) - \left[\left(\frac{3}{2}M - dM\right)\left(v + dv\right) - vdM\right]}$$ Simplifying $$\displaystyle{dp = -\frac{3}{2}Mdv + 2vdM$$ Dividing by $$dt$$ $$\displaystyle{\frac{dp}{dt} = -\frac{3}{2}M\frac{dv}{dt} + 2v\frac{dM}{dt}}$$ As $$\displaystyle{\frac{dM}{dt} = -k}$$ $$\displaystyle{\frac{dv}{dt} = -\frac{4}{3M}vk - \frac{2F}{3M}}$$ Im confused because of the very negative right hand side of the equation. Did i make an error in the set up at the start? Thankyou in advance