Essential Physics

Rocket propulsion


The puzzle of propulsion	Many people think that rockets work by the propellant pushing against the air or the launch pad. If this were true, then how would rockets still work in outer space? There is nothing in the vacuum of outer space to push against! Space travel is made possible by the physics of momentum conservation.
The need for propulsion	You might think that a single impulse from the launch pad would suffice to get a rocket to the Moon, Mars, or elsewhere. After all, an object in motion remains in motion unless acted upon by an outside force; and since there isn’t any air resistance in outer space, one might guess that a rocket could coast to its destination. At first glance, that argument sounds reasonable. Nevertheless, a space-faring rocket needs to be able to push itself forward, backward, or sideways at many points in its journey. For starters, Earth’s gravity extends beyond the atmosphere, and any spacecraft headed to Mars, Jupiter, or Saturn has to fight Earth’s gravity as well as the Sun’s. Furthermore, spacecraft need to maneuver to go into orbit around—or land on—distant moons, planets, or asteroids.
Action– reaction forces in rocketry	Rockets are propelled “forward” when exhaust gases stream out “backward.” Since those gases have mass, they have momentum—momentum that they lacked until they were blasted out of the exhaust nozzle. So the rocket has a forward (or positive) momentum while the exhaust has a backward (or negative) momentum. Since the rocket and exhaust form a closed system, their total momentum must be conserved.
Impulse in rocket propulsion	If the change in momentum is zero, then the total impulses must be zero because impulse is the change in momentum. The exhaust gases were given a negative (or backward) impulse, while the rocket itself received a positive impulse because of the law of conservation of momentum. And that means you can push a rocket forward as long as you have something (such as exhaust gases) to toss in the opposite direction—no launch pad, atmosphere, or other obstacle is required.
A note on sign conventions	The previous paragraph and figure may imply that rockets always move in the positive x-direction. But that’s just the convention we adopted here for convenience. We could just as well have said that the exhaust gases travel in the positive x-direction—or the positive y direction, for that matter. As long as you are consistent in assigning plus (+) and minus (−) signs to action–reaction forces and the impulses they cause, you should be able to correctly solve any problems involving propulsion.

Increasing acceleration	A typical, modern rocket experiences an increasing, not constant, acceleration. Why? In a rocket, the force of thrust (which is the product of the exhaust velocity and the mass-loss rate, measured in newtons) is more or less constant until the fuel is nearly used up. The mass of the rocket decreases over time as the fuel is used up and expelled out the nozzle. Now combine these two ideas: A constant force divided by a decreasing mass implies an increasing acceleration from Newton’s second law, a = F/m. That is exactly what we see in the first few minutes of a rocket launch. Not only does the rocket gain speed, it does so at an increasing rate.
Two-stage systems	A rocket gets more “bang” (acceleration) for the “buck” (thrust) as it loses mass—as fuel is consumed, there is less and less mass to push forward. Could part of the rocket be cast off as well? The answer is yes, and doing so makes rocketry more efficient. Two-stage systems divide the propellant between two separate engines, and the first is discarded (typically to land in the ocean) when its fuel is spent. This means that no more work has to be done lifting many tons of empty hardware.
Parameters for modern rockets	A modern cargo-carrying rocket might have a mass of 100,000 kg when empty and 500,000 kg on the launch pad, full of the fuel it will use for the first stage of its journey. Exhaust typically exits such rockets at many thousands of meters per second, with hundreds or thousands of kilograms flowing out each second.
Test your knowledge	Suppose that a single-stage rocket can provide a thrust of 10⁷ N. When empty, it has a mass of 100,000 kg; it can carry 400,000 kg of fuel. What will its acceleration be when its fuel tank is half-empty? (Assume that air resistance can be neglected and that the acceleration due to gravity can be approximated as g = 10 m/s².) State your answer in both meters per second squared and in “g’s.” Note: You may wish to construct a free-body diagram similar to that shown here. Be careful: Observe that this diagram shows the mass, weight, and net force for a full fuel tank. Yours should reflect the mass, weight, and net force that correspond to half as much fuel.
	Answer: The rocket undergoes an instantaneous acceleration of 23 m/s², or 2.3g. Asked: acceleration a in meters per second square and in terms of g Given: mass m = 100,000 kg plus half of 400,000 kg, thrust force = 1×10⁷ newtons, g = 10 m/s² Relationships: a = F/m; weight = mg; net force = thrust − weight Solve: First calculate the total mass of the rocket plus remaining fuel: $m = 100, 000 kg + \frac{1}{2} (400, 000 kg) = 300, 000 kg$ Next, calculate the rocket’s weight: $F_{w e i g h t} = m g = (300, 000 kg) (10 {m/s}^{2}) = 3 \times 10^{6} N$ Now assess the net force and apply Newton’s second law: $F_{n e t} = F_{t h r u s t} + F_{w e i g h t} = 1 \times 10^{7} N + (3 \times 10^{6} N) = 7 \times 10^{6} N$ $a = F_{n e t} / m = (7 \times 10^{6} N) / (3 \times 10^{5} kg) = 23 {m/s}^{2}$ The rocket undergoes an instantaneous acceleration of 23 m/s², or 2.3g.


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