A-level Physics (Advancing Physics)/Rockets, Hoses and Machine Guns

We have already seen that force is the rate of change of momentum. This applies to continuous flows of momentum as well as to collisions:

$$F = \frac{dp}{dt} = v\frac{dm}{dt}$$

If I have a machine gun, explosions give the bullets of mass m momentum, causing them to move at a velocity v. This occurs several times each second - the momentum of the bullets is changing, and so there is a roughly continuous force acting on them. Momentum, of course, must be conserved. This results in a change in the momentum of the gun each time it fires a bullet. Overall, this results in a roughly continuous force on the gun which is equal and opposite to the force acting on the bullets.

If I have a tank of water and a hose, with a pump, and I pump the water out of the tank, a similar thing occurs - a force pushes me away from the direction of flow of the water. This force is equal to the flow rate (in kgs−1) of the water multiplied by its velocity. Bear in mind that 1 litre of water has a mass of about 1 kg.

Rockets work on this principle - they pump out fuel, causing it to gain momentum. This results in a thrust on the rocket. When designing propulsion systems for rockets, the aim is to give the fuel as high a velocity per. unit mass as possible in order to make the system fuel-efficient, and to get a high enough change in momentum.

Questions
1. A machine gun fires 300 5g bullets per. minute at 800ms−1. What force is exerted on the gun?

2. 1 litre of water is pumped out of a tank in 5 seconds through a hose. If a 2N force is exerted on the tank, at what speed does the water leave the hose?

3. If the hose were connected to the mains, what problems would there be with the above formula?

4. The thrust of the first stage of a Saturn V rocket is 34 MN, using 131000 kg of solid fuel in 168 seconds. At what velocity does the fuel leave the tank?

5. Escape velocity from the Earth is 11 km−1. What is the velocity of the rocket after the first stage is used up, if the total mass of the rocket is 3 x 106 kg? How does this compare to escape velocity?

/Worked Solutions/