Mass Ratio - Derivation

Derivation

The definition arises naturally from the Tsiolkovsky's rocket equation:

where

• Δv is the desired change in the rocket's velocity
• v_e is the effective exhaust velocity (see specific impulse)
• m₀ is the initial mass (rocket plus contents plus propellant)
• m₁ is the final mass (rocket plus contents)

This equation can be rewritten in the following equivalent form:

The fraction on the left-hand side of this equation is the rocket's mass ratio by definition.

This equation indicates that a Δv of times the exhaust velocity requires a mass ratio of . For instance, for a vehicle to achieve a of 2.5 times its exhaust velocity would require a mass ratio of (approximately 12.2). One could say that a "velocity ratio" of requires a mass ratio of .

Sutton defines the mass ratio inversely as:

In this case, the values for mass fraction are always less than 1.