### References

Understanding Space: An Introduction to Astronautics, Revised Edition by Jerry Jon Sellers

### Tsiolkovsky's Rocket Equation

One of the most important equations you will encounter in rocketry is Konstantin Tsiolkovsky's “Rocket Equation”; given below.

Δv = VE * ln(ML / ME)

Where:

Δv = Final velocity (Delta-vee or Δv) of the rocket in meters per second or feet per second.

VE: Velocity of the rocket's exhaust in meters per second or feet per second.

ML: Total mass of the rocket fully loaded (with payload, propellant, etc).

ME: Empty mass of the rocket at burnout with all propellant expended.

The Mass Equation

Mb and Me can be in pounds, kilograms, grams, or tons as long as both use the same unit!

### Total Rocket Thrust Equation

FThrust = (ṁ * Vexit) + Aexit * (PExit – PAtm)

Where

FThrust = Thrust in Newtons (N)
= Mass flow rate in kilograms per second (kg/s)
Vexit = Exit velocity of exhaust in m/sec
Aexit = Exit Area of Nozzle in square meters
PExit = Exit Pressure in N/m2 (Pascals)
PAtm = Atmospheric Pressure in N/m2 (Pascals)

PF = T / VE

### Equation Explained

PF = Propellant flow in kg/sec.

T = Thrust in Newtons

VE = Exhaust Velocity of Engine in m/sec.

 EXAMPLE: The Space Shuttle Main Engine (SSME) has a ISP of 452 seconds (4,434.12 m/sec exhaust velocity) in vacuum and a thrust of 2.18 meganewtons. What is the propellant flow in kg/sec? 2,180,000 N / 4,434.12 m/sec = 491.64 kg/sec PF.

# ISP Equations

### Computing ISP from Actual Engine Performance:

ISP = (Thrust * Time) / MProp

Where:

ISP = Specific Impulse of the Propellant/Engine System
Thrust = Thrust of the engine in force.
Time = Length of time the engine operates in seconds.
MProp = Mass of propellant consumed by the engine during it's operation
Note: All units must be internally consistent; e.g. if you use pounds of thrust, you must also measure propellant consumption in pounds.
 EXAMPLE: We have a solid rocket motor which burns for 15 seconds, producing 24,000 lbs of thrust; consuming 1,400 lbs of propellant in the process. (24,000 * 15) / 1,400 = 257.14 ISP

### Computing ISP from Exhaust Velocity:

ISP = VE / 9.81 m/sec2 (Metric)

ISP = VE / 32.2 ft/sec2 (Standard)

Where:

ISP = Specific Impulse
VE = Exhaust Velocity (m/sec or feet/sec)
 EXAMPLE: We have an engine that has an exhaust velocity of 8,093 meters per second. What is it's ISP? 8,093 / 9.81 = 824.97 ISP

### Computing Exhaust Velocity from ISP:

VE = ISP * 9.81 m/sec2 (metric)

VE = ISP * 32.2 feet/sec2 (Standard)

Where:

VE = Exhaust Velocity of the engine in meters per second or feet per second.

ISP = Specific Impulse of the engine.

 EXAMPLE: We have an engine that has an ISP of 400. What is its exhaust velocity in m/sec? 400 * 9.81 = 3,924 m/sec Exhaust Velocity

# Misc. Delta-X Equations

### Computing Delta-T (Length of Burn)

ΔT = [ (ML * EV) / F ] * [1 – EXP[-(ΔV / EV)]

Where:

ΔT = Length of burn in seconds

ML: Total mass of the rocket at the beginning of the burn.

EV = Exhaust Velocity in meters/second.

F: Thrust of the rocket in Newtons.

ΔV = Delta-V of burn in meters/second.

(Equation in Latex Format)

 EXAMPLE: The Apollo 11 CSM Columbia is executing her Trans-Earth-Injection (TEI) Burn to return to earth with the following key parameters: Velocity Change (ΔV): 3,279 fps (999.444 m/sec)Spacecraft Mass: 36,965.7 lbs (16,767.35 kg)Spacecraft Thrust: 20,500 lbf (91,200 newtons)Spacecraft ISP: 314 seconds (3,080.34 m/sec exhaust velocity) How long will the TEI burn take? [ (16,767.35 * 3080.34) / 91200 ] * [1 – EXP[-(999.444 / 3080.34)] = 156.9192 seconds (Solution in Latex Format) This compares favorably with the actual 151.4 second burn that Apollo 11 executed to return to Earth.

### Computing Delta-P (Amount of Propellant Consumed in a Burn)

ΔP = ML – [ ML / EXP( ΔV / EV ) ]

Where:

ΔP = Amount of propellant consumed in kilograms.

ML: Total mass of the rocket at the beginning of the burn.

ΔV = Delta-V of burn in meters/second.

EV = Exhaust Velocity in meters/second.

(Equation in Latex Format)

 EXAMPLE: The Apollo 11 CSM Columbia is executing her Trans-Earth-Injection (TEI) Burn to return to earth with the following key parameters: Velocity Change (ΔV): 3,279 fps (999.444 m/sec)Spacecraft Mass: 36,965.7 lbs (16,767.35 kg)Spacecraft ISP: 314 seconds (3,080.34 m/sec exhaust velocity) How much propellant will the TEI burn consume? 16767.35 – [ 16767.35 / EXP( 999.444 / 3080.34 ) ] = 4645.925 kilograms (Solution in Latex Format) This compares very favorably with Apollo 11’s TEI Burn, which consumed 10,173 lbs (4,614.4 kg) of propellant, leaving the CSM with a end of burn mass of 26,792.7 lbs (12,153 kg).