Rain falling into a moving cart

If 1 kg/s of rain falls vertically into a 10-kg cart that is rolling without friction at an initial speed of 1.0 m/s, what is the effect on the speed of the cart when the rain first starts falling?

The rain and the cart make horizontal forces on each other, but there is no external horizontal force on the rain-plus-cart system, so the horizontal motion obeys

We use the product rule to find

We are trying to find how v changes, so we solve for dv/dt,

(This is only at the moment when the rain starts to fall.)