One of the most basic and often used rocket equations is called the "ideal rocket equation." It relates exhaust speed and mass ratio to the resulting change in speed, Dv, of the rocket. Last section, exhaust speed and mass ratio were discussed in terms of momentum. Here, we derive the rocket equation by starting with the force equation from section 2.3, and using c for the effective exhaust speed,

For a rocket, reactive force is due entirely to its change in mass by expelling exhaust gas at speed, c. With no external (applied) force acting on the rocket, F = 0, and the above equation becomes:

By dividing this equation through by rocket mass, m, and multiplying each side by dt,

This differential equation can be solved by integral calculus. The result is the ideal rocket equation:

where Dv is the change in rocket speed, "ln" is the natural logarithm and MR is its mass ratio. To maximize speed, both rocket effective exhaust speed, c, and mass ratio, MR, are maximized. The effective exhaust speed relates to the propulsion system performance, while the mass ratio is a figure of merit of the structural design. The propulsion designer tries to maximize I_sp (or c) and the structural designer tries to maximize MR.