Hi I have this problem involving a cart which is losing sand(adsbygoogle = window.adsbygoogle || []).push({});

It says:

A cart with initial mass [tex]M[/tex] and a load of sand [tex]\frac{1}{2}M[/tex] loses sand at the rate [tex]k[/tex] kg/s. The cart is pulled horizontally by a force [tex]F[/tex]. Find the differential equation for the rate of change of the carts velocity in terms of [tex]k,M[/tex] and [tex]F[/tex] while there is sand in the cart.

So i said that at [tex]t = 0[/tex] the momentum[tex] = \frac{3}{2}Mv[/tex]. Therefore

[tex]\displaystyle{dp = \left(\frac{3}{2}Mv\right) - \left[\left(\frac{3}{2}M - dM\right)\left(v + dv\right) - vdM\right]}[/tex]

Simplifying

[tex]\displaystyle{dp = -\frac{3}{2}Mdv + 2vdM[/tex]

Dividing by [tex]dt[/tex]

[tex]\displaystyle{\frac{dp}{dt} = -\frac{3}{2}M\frac{dv}{dt} + 2v\frac{dM}{dt}}[/tex]

As [tex]\displaystyle{\frac{dM}{dt} = -k}[/tex]

[tex]\displaystyle{\frac{dv}{dt} = -\frac{4}{3M}vk - \frac{2F}{3M}}[/tex]

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

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# A Cart losing Mass

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