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

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

$$F = v\frac{dm}{dt} = 800 \times \frac{300 \times 0.005}{60} = 20\mbox{ N}$$

'''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?'''

The flow rate is about 0.2 kg s−1.

$$2 = 0.2v$$

$$v = \frac{2}{0.2} = 10\mbox{ ms}^{-1}$$

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

The force would not be exerted on the tank, but would instead be referred back through the pipe - it is not gaining velocity at the nozzle.

'''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?'''

$$34 \times 10^6 = v\frac{131000}{168}$$

$$v = \frac{34 \times 10^6 \times 168}{131000} = 43600\mbox{ ms}^{-1}$$

'''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?'''

$$ma = 34 \times 10^6 - mg$$

$$a = \frac{34 \times 10^6}{3 \times 10^6} - 9.81 = 1.52\mbox{ ms}^{-2}$$

$$v = at = 168 \times 1.52 = 255.92\mbox{ ms}^{-1} \approx 0.256\mbox{ kms}^{-1}$$

This is 0.002% of escape velocity, hence the need for the other stages of the rocket.