1. MAE 4262: ROCKETS AND MISSION ANALYSIS
Rocket Equation and Losses
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. ROCKET EQUATION: IMPORTANT TRENDS

3. TYPICAL DV MISSION REQUIREMENTS

4. DV CAPABILITY FOR VARIOUS ROCKETS
REF: Space Propulsion Analysis and Design, by Humble, Henry and Larson

5. TYPICAL PERFORMANCE PARAMETERS (T and Isp)

6. GRAVITY
Remember that gravity on Earth (~ 9.81 m/s2) may be calculated fundamentally
Average radius of the Earth ~ 6,378 km or 3,963 miles
Mass of the Earth ~ x1024 kg
Some typical values for Earth:
High power amateur model rocket ~ 100,000 ft, 30.5 km, 19 miles
g/ge = 99%
Shuttle in LEO (altitude of 300 km, 186 miles)
g/ge = 91%
Satellite in GEO (altitude of 42,000 km, 26,000 miles)
g/ge = 1.7%
Note that the radius of the moon is about 1,737 km and mass is 7.36x1022 kg
So g on the surface of the moon is about 1.62 m/s2

7. COMPARISON OF GRAVITY, DENSITY AND PRESSURE VERSUS ALTITUDE

8. COMPARISON OF GRAVITY, DENSITY AND PRESSURE VERSUS ALTITUDE
Shuttle
LEO

9. TYPICAL DRAG VARIATION FOR ROCKETS

10. Variation of lift and drag coefficient with Mach number of V-2 rocket missile based on body cross-sectional area with jet off

11. DRAG: SUPERSONIC MISSILE EXAMPLE

12. COMMENTS: LAUNCH FROM SURFACE OF EARTH
To get to orbit (or to escape), direction of travel must be parallel to Earth’s surface (not perpendicular)
We launch vertically off the surface of the Earth, WHY?
Gravity
When rocket is vertical, gravity is acting against T and V
Drag
V2 dependence: Drag ↑ as rocket accelerates
Large effect in lower atmosphere
Acceleration of vehicle is almost constant even though mass is changing
Density dependence: r ↓ very rapidly in atmosphere (r/rS.L. ~ 1% at 100,000 ft)
All rocket pass through condition of maximum dynamic pressure (MAX Q)
Many rockets stay vertical through this part
Get through atmosphere as quickly as possible
BUT before rocket really starts to speed up

13. COMMENTS: LAUNCHERS
Need certain velocities to get to space (and stay in space), escape, insertion, transition velocities, etc. → give DV requirements
Don’t want to carry fuel (heavy fuel is working against you)
Burn fuel early in flight → high accelerations, V2 ↑
Atmosphere is counter argument: drag, dynamic pressure
Why not launch horizontal?
Less gravity loss
Drag loss is high, more time in atmosphere
Lots of structural stress
Launch might look different on moon
Vertical launch segment:
Get out of dense atmosphere quickly, but still at relatively low speed
Don’t spend too much time here (vertical segment contributes nothing to eventual vertical orbital velocity)
Highest gravity losses, but sustain them to get lower density then really increase DV

14. Variation in air density (r), velocity (V), altitude (h), and dynamic pressure (q) during a Space Shuttle launch