Hosepipes, helicopters and conveyor belts

1. Hosepipe

You can appreciate that if you are hit by a fast moving stream of water from a hose pipe you experience a force - if this hosepipe is a water cannon then the force could be great enough to knock you over.

Consider a hosepipe of cross sectional area A that gives a jet of water of velocity v. If this hits a vertical wall and loses all its momentum then in one second a jet of length v will hit the wall and so:
Mass of water striking the wall per second = rvA where r is the density of the water

Force on wall = rate of change of momentum of water = change of momentum per second = [rvA]v = rv²A

Example problem
A water cannon has a cross sectional area of 25 cm² (25x10^-4 m²). It ejects a jet of water (density 1000 kgm^-3) at a speed of 20 ms^-1.
Calculate the force in the jet.

Force = rAv² = 1000x25x10^-4x20² = 1000 N

2. The hovering helicopter

Imagine a helicopter hovering above the ground. The blades on the rotor are angled slightly so that as they spin round they thrust a column of air downwards - it is this column of air moving vertically that keeps the helicopter up. The faster the blades rotate and the larger they are the bigger the lift, so large helicopters need large blades or even two rotors.

Now as before for the water jet force = rate of change of momentum and so for the helicopter to hover the momentum change per second of the air column must be equal to the weight of the helicopter.

Using both Newton's second and third laws:

mg = rpr²v²

so for a helicopter of mass 4500 kg, air of density 1.2 kgm^-3 and rotor blades 6.2 m in radius the vertical speed of the air column produced by the rotor blades must be 17 ms^-1.

The power of the helicopter

Remember that Power = Force x Velocity however in this case the velocity of the air column is being increased from zero to v and so the average velocity is v/2. This gives the power as:
Power = Force x average velocity = rpr²v² x v/2 = rpr²v³
Some examples of powers: Lynx helicopter power range 670 kW - 835 kW Power boat = 112 kW

3. Conveyor belt and moving walkway

(This description could apply either to a person stepping onto a horizontal moving walkway or to sand being tipped onto a conveyor belt)


Imagine that people of total mass m step onto a moving walkway every second. If the velocity of the walkway is v then:
gain in momentum per second = mv

but this is the rate of change of momentum and therefore the force needed to move the belt is:

Force in conveyor belt :- space Force = mv