First, read How Rocket Engines Work to get the basic ideas.

### The Rocket Equation

Every rocket launched, from little model rockets to the huge Saturn V rockets and the Space Shuttle launch rockets, must abide by the Rocket Equation. This equation and variations are used predict how fast a rocket will travel (velocity) and how high it will fly (altitude).

 Rocket Equation describing motion at time t: Rocket Equation taking into account external forces (For example, a rocket taking off vertically) Rocket Equation solution: KEY (note the use of metric units) M= total rocket mass (kg) dv/dt = change in velocity at time t uex = exhaust velocity (m/s) dM/dt = change in rocket mass at time t g = gravity (assumed constant at 9.8 m/s2 for launches from Earth's surface) t = time (s)

Despite the fact that a lot of relatively complicated math is involved, the equation above ignores a lot of details, like the effects of air resistance (drag), the weight of the rocket body, and changing gravity and air density as the rocket ascends. I suspect this is due to the relative scale of the numbers involved, since the equations above accurately characterize large-scale launches like the Space Shuttle and the Saturn V rocket.

### A Simple Example

 Saturn V Total Mass: 2.5 * 10^6 kg Exhaust velocity: 3.0 km/s (3000 m/s) Burn Rate: 1.6 * 10^4 kg/s Duration: 2 min (120 sec) Space Shuttle (for contrast) Total Mass: 2 * 10^6 kg Exhaust velocity: 6000 mph (2681.6 m/s) 2 SRBs Thrust: 1.5* 10^6 kg each SRB Burn Duration: 2 minutes (120 sec) 3 Main Engines Thrust: 107400 kg each Main Engine Burn Duration: 8 minutes (480 sec

If we take the equation above, we can calculate the velocity of a Saturn V rocket just as the burn ends. We need to pre-calculate the mass of fuel expended by multiplying the burn rate times the duration to get 1.92 * 10^6 kg. Then, just substitute in the numbers for the Saturn V as shown below.

v(120) = (3000 m/s) * ln (2.5 * 10^6 kg/[2.5 * 10^6 kg - 1.92 * 10^6 kg] ) - 1176 m/s

v(120) = 3207 m/s

Graph of rocket velocity during the burn.

Experiment with the Rocket Equation (note: requires Java-enabled browser)

### Multi-Stage Rockets

Multi-stage rockets are more often used for large-scale launches. The Rocket Equation needs a small amount of alteration to take into account multiple burns and diminishing total mass due to discarding used-up rocket stages and fuel for each stage. The important thing to remember is that the second following stages of a multi-stage rocket start acceleration from the final speed of the previous stage. Essentially, you're adding the velocities produced by each stage.

When the first stage ends at time t1, the fuel (m1) has been burned. Then the first stage rocket casing is dropped (m'1), so the starting mass for the second stage is (m0 - m1 - m'1). The final mass is (m0 - m1 - m'1 - m2), where m2 is the mass of second stage fuel burned after additional time t2. All those manipulations produce the following equation:

This is a generalized form of the Rocket Equation.

Send a payload of 2 kilograms to an altitude of 200 kilometers on or before 08 Nov 2000, using a privately-developed launcher.
http://www.space-frontier.org/Events/CATSPRIZE_1/

MIT Rocket Team for the CATS contest.
http://web.mit.edu/cats/

Explanations of Rocket Motion