What is the mass of water ejected per unit time from the nozzle in a toy rocket?

[gaobo9109]

Homework Statement

A toy rocket consists of a plastic bottle which is partially filled with water. The space above the water contains compressed air. At one instant during the flight of the rocket, water of density p is forced through the nozzle of radius r at speed v relative to the nozzle. Determine, in terms of p, r and v, the mass of water ejected per unit time from the nozzle

Homework Equations

The Attempt at a Solution

I know the answer is πr²pv. But I am not sure of exact explanation why this is the case.

 

[schmeling]

its not so much about momentum but really about understanding the geometry of the problem.

You know that the water has a certain velocity, you know how far a single particle of water will travell in one second. You are also given the radius of the nozzle therefore you can easily figure out the volume of the water that leaves the bottle every second.
It will be the area of the nozzle multiplied by the distance traveled by the water in one second ( that is V=pi*R^2*v)

once you have the volume its all easy as you can use the density to get the mass. it's just the volume multiplied by the density.