Rocket Thrust

Rocket force (thrust) can be found by the conservation of momentum principle. The trick is to include the exhaust as part of the system. The rocket pushes backward on the exhaust and therefore receives a forward force. Assume no other forces act (we can deal with external forces later), so momentum is conserved. If you have not studied calculus, think of d as a very small D, so dv is a small change of v.

The rocket is losing mass as the exhaust spews out, so the mass of the rocket is a decreasing function of time: dm/dt is negative if m is the mass of the rocket (including contents). In what follows, I will make dm the (positive) mass of exhaust that comes out in time dt, so the rate of change of mass of the rocket is -dm/dt, not dm/dt.

Let v_rel be the velocity of the exhaust relative to the rocket, and define the rocket's v as positive (see sketch below), so v_rel is negative. After leaving the rocket, the exhaust has velocity v + v_rel , which could be to the right, smaller than v, or to the left.

Initially the rocket had mass m and velocity v; time dt later (shown below) it has m-dm and v+dv.

v + dv --->

Now write the conservation of momentum:

initial momentum = momentum of both at time dt later
mv = (m-dm)(v+dv) + dm(v+v_rel)

Do the algebra and note dmdv ≈ 0, and divide by dt & show that

-v_reldm/dt = mdv/dt. But that is ma on the right, or the force on the rocket. (Remember that v_rel is negative, so we have a positive force. Or if you prefer,
F = v_reldm/dt with everything now defined to be positive. Check the units, and remember that a newton is a kg∙m/s².
The above equation applies to more than rockets: a balloon with the air rushing out, the force on the hose when firefighters are putting out a fire, the average force on a machine gun…. A reverse rocket: railroad car moving with speed v under a grain chute from a grain elevator, and the grain is coming straight down with mass rate dm/dt. This causes a force against its motion = vdm/dt. Another reverse rocket: intake of a jet:

Jet Engines

Taking in air in the front causes a backward force on the engine. The intake acts as a rocket in reverse. So the equation for the net thrust is the same as above except that you need to subtract [speed of the plane with respect to the air times the mass of air intake/time]. It takes in less mass than it shoots out, because the burned fuel adds to the output mass, and it shoots out much faster than the intake airspeed.

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