Abox sliding on a frictionless surface when it suddenly begins to rain heavily. if the downpour of rainfall falls vertically into the box, gradually filling it, what would happen to the speed of the box

The speed of the box will decrease with the downpour of rainfall into the box.

Explanation:

The box when sliding on a friction-less surface will exhibit momentum p = mass * velocity. The moment the rain drop falls on the box, collision of rain drops with the box occurs leading the raindrop to collide and stick to the box.

According to law of conservation of momentum, the total momentum of objects before collision should be equal to the total momentum of the objects after collision.  

If we consider M as mass of the box and m as mass of the rain drop. The initial and final velocities of box are considered as U1 = 0 and V1 = Vb, respectively.

After collision of rain drop with the box or filling of rain drop in the box, the mass of the box will be sum of mass of box and rain drop, i.e., (M+m) and the velocity is unknown to us.

According to the law of conservation of momentum equation it can be seen the speed after filling of raindrops in the box (V) is inversely proportional to the sum of the masses of the box and raindrops, so as the raindrops increases in the box, the mass of raindrops (m) will be increasing.

This will lead to increase of the value (m+M), which means that the speed of the box will decrease on increase in the filling of raindrops in the box.