Rocket Equation,
Dynamics, and
Launching
Ken Chen
COSMOS 2013
C3 Engineering Mechanics
Why Rockets?
• Imagine a world where you didn’t have these:
–
–
–
–
Cell phones
Television
Internet
GPS Navigation
• Rockets are used for:
– Space transportation/exploration
– Satellite communications
– Military weaponry
Thank rockets!
Parts of a Rocket
http://exploration.grc.nasa.gov/education/rocket/Images/rockpart
if
Forces on a Rocket
• On Center of Gravity
(CG)
– Weight
– Thrust (mostly)
• On Center of Pressure
(CP) (Aerodynamic)
– Lift
– Drag
http://exploration.grc.nasa.gov/education/rocket/rktaero.ht
l
Thrust
• Thrust-Newton’s third law
– Rocket pushes on propellant, propellant pushes back
on rocket
– Thrust equation:
F
thrust
 m  ve  ( pe  p0 )  Ae
• Ways to increase thrust:
– Maximize mass flow rate
– Maximize exit velocity to incoming velocity
• Correction term ( p  p )  A
e
0
e
– Pressure difference
Drag
• Frictional drag
– Caused by friction of fluid against the surface
of the body
1 2
• Pressure/Form drag
D  v AC D
2
– Different shapes and sizes
– Vortices in the wake of the body
• Wave drag
– Common at transonic and supersonic speeds
– Can reach up to 4x subsonic drag
Control
• CP below CG –
stable equilibrium
• Gimbaled Thrust
– Nozzle can be
shifted/rotated;
changes direction
of thrust
• Vernier Rocket
– Small additional
rocket engines
http://exploration.grc.nasa.gov/education/rocket/rktcontrl.html
Konstantin E. Tsiolkovsky (18571935)
• Early education
impaired by
deafness
• College – Moscow
• Studied
mathematics and
science
– Monoplane
– Wind tunnel
http://www.allstar.fiu.edu/aero/images/Tsiolkovsky.jpg
Tsiolkovsky’s Contributions
• Rocket equation
– Still remains the basis of all rocket dynamics
today
• Founder of rocket dynamics
– Understood dynamics of firework rockets
through math and physics
• Anthropocosmism
– Humans will dominate space
• Inspired several other great minds in
rocketry
Robert H. Goddard (1882-1945)
• Masterful physicist
and inventor
• Built/tested first
liquid fuel rocket in
1926
• Patented multi-stage
rocket in 1914
• German V2 missiles
– Steering, vanes
http://www.rugusavay.com/wp-content/uploads/2013/03/Robert-Goddard-4.jpg
Tsiolkovsky Rocket Equation
 m0
V  Veq  ln
m
 f
Derivation:
– Conservation of momentum
dp
F
0
dt




dp  p(t  dt )  p (t )
m
v
p(t )  m  v
(v  dv)  ve
dm
m  dm
p(t  dt )  (m  dm)  (v  dv)  dm  (v  dv  ve )
dp  (m  dm)  (v  dv)  dm  (v  dv  ve )  mv
dp  mv  mv  vdm  vdm  dmdv  dmdv  mdv  ve dm
dp  mdv  ve dm  0
dp  mdv  ve dm  0
v  dv
Flip sign
Tsiolkovsky Rocket Equation (cont.)
A parallel:
dp  mdv  ve dm  0
v
dv   e  dm
m
Newton’s Second Law
1
dv


v

 dm
e 
v0
m0 m
v f  v0  ve  (ln(m f )  ln(m0 ))
vf
mf
 m0
v  ve ln
m
 f
Ideal Rocket
Equation




•Constant Mass
Fnet  m  a
•Variable Mass
Fnet 
d (mv)
dv
dm
 m  v
dt
dt
dt
Specific Impulse Isp
• A way to measure engine efficiency
F
I sp 
m  g
• Can also be used to determine thrust
What actually happens -
Launching
• Potentially dangerous
– Astronauts bear about 3 g’s of acceleration
– Requires a high amount of precision
– Sound energy can cause damage
• Liftoff process
– Entire process controlled by computers
– Increase in air pressure – overstressing
– Need enough thrust
Future Rocket Dynamics
• Nuclear Propulsion
– Abandoned in 1972 (Project Rover)
– Comeback?
• ANTIMATTER Propulsion
– Involves the annihilation of
matter
– Efficiency is incomparable –
interstellar missions
http://static.ddmcdn.com/gif/antimatter-1.jpg
More on Antimatter Propulsion
• When combining electron with anti-electron


– Photons (gamma rays) e  e  2
• Proton with anti-proton


o
– Pions p  p  n  n  n
• Decay into muons and neutrinos
• Eventually all annihilate or decay into massless particles,
photons, and neutrinos
• Facts
– Exhaust velocity ~0.95 c
– Storage and production of antiprotons
• 1 ng/year
• Comparison
– Apollo 11 took 4 days to reach the moon
– In 4 days, a pion antimatter rocket can travel from Earth to
Sun 350 times
References
•
•
•
•
•
•
•
•
"Basic Rocket Motion." Guided Tours. Ed. Tom Benson. NASA, n.d. Web. 29
July 2013.
Braeunig, Robert A. "Basics of Space Flight: Rocket Propulsion." Rocket
and Space Technology. N.p., 2012. Web. 29 July 2013.
Brewster, Isaac. "The Physics Behind The Rocket." The Physics of Rockets.
University of Alaska Fairbanks, 31 Oct. 2000. Web. 29 July 2013.
Dunbar, Brian. "Konstantin E. Tsiolkovsky." NASA. NASA, 20 Sept. 2010.
Web. 01 Aug. 2013.
Dunbar, Brian. "Societal Impact of the Space Age." NASA. NASA, 07 Nov.
2005. Web. 29 July 2013.
Dunbar, Brian. "What Is a Rocket?" NASA. NASA, 12 July 2011. Web. 29
July 2013.
Garner, Rob. "Robert H. Goddard." NASA. NASA, 28 July 2013. Web. 01
Aug. 2013.
Hafez, Mohamed M. "COSMOS Engineering Mechanics." COSMOS 2013.
Davis, California. July 2013. Lecture.
References (cont.)
•
•
•
•
•
•
•
Keith, Edward L. "Fundamentals of Rockets and Missiles." Laurel,
Maryland. Lecture.
Krempetz, Kurt. "Freeflight Trimming." AMA Glider. Academy of Model
Aeronautics, May 2006. Web. 29 July 2013.
Portland State Aerospace Society. "Simple Rocket Science." Simplerockets
1d. Portland State University, 02 Mar. 2010. Web. 29 July 2013.
Qualitative Reasoning Group. "Propulsion." How Are Rockets
Designed? Northwestern University, n.d. Web. 29 July 2013.
Turner, Martin J. L. Rocket and Spacecraft Propulsion: Principles, Practice
and New Developments. Berlin: Springer, 2009. Print.
Walter, Ulrich. Astronautics. Weinheim: Wiley-VCH, 2008. Print.
Wilkinson, Peter. "Rocket Launch and Reentry." Senior Science. St. Francis
Xavier's College, Hamilton, n.d. Web.