The Mathematics of Rocket
Propulsion
BY:
Ben
Ferguson
Abhishek Gupta
Matt Kwan
Joel Miller
Rocketry in the Contemporary
Age

Robert H. Goddard

Werner Von Braun and the V-2 Rocket

NASA

Military Applications

Amateur Rocketry
Mathematical Relationships
Critical to Understanding
Rocket Propulsion

Impulse
 Velocity
 Acceleration
Impulse
The impulse of a force is a product of a force and the timeframe in which it acts.
Impulse is given by the integral:
If a constant net force is present, impulse is equal to the average impulse:
Remember that impulse is not a force or event, but a physical quantity.
As such, it is often idealized for use in predicting the effects of ideal collisions as well as ideal engine output in rockets.
Velocity/Acceleration
Velocity is a measure of the rate of change in an object’s displacement from a certain point.
Velocity is given in units of distance per unit time:
Acceleration is a measure of the rate of change in an object’s velocity,
or the derivative of the velocity function evaluated for a certain time ‘t’:
Acceleration is expressed in units of distance over units of time squared:
Ex: m/s^2
The kinetic energy of any object is defined as:
Where m is the mass of the object and v is the velocity at time ‘t’
Finding The Acceleration
of a Rocket

Pi=Pf
 Pi=Mv , Pf= -dMUp + (dM+M)( v+dv); Where v
Use Conservation of Momentum
is velocity of rocket, Up is velocity of propellant, and M is mass of rocket


U =(v+dv)-up ; Where up is velocity of propellant
Substitute
p
relative to the rocket
Mv= -dM(v+dv-up) + (dM+M)(v+dv) then use
the distributive property

Mv= -dM(-up) -dM(v+dv) + dM(v+dv) +
M(v+dv)
Finding The Acceleration
of a Rocket

Mv= -dM(-up) -dM(v+dv) + dM(v+dv) +
M(v+dv)
 Mv= -dM(-up) + M(v+dv)
 Mv= dMup + Mv + Mdv
 0= dMup + Mdv
 -dMup= Mdv divide both sides by dt
 -dM/dt up =Mdv/dt
 -dM/dt is rate of fuel consumption and dv/dt is acceleration a
 -dM/dt up is known as thrust T so…
 T=Ma
Finding the Velocity
-dMup= Mdv
 -dM/M up= dv integrate

Remember that
divide both sides by
M
 ∫-up M-1dM = ∫dv; from Mi to Mf and vi to vf

-up (lnMf -lnMi) = vf -vi
 up(lnMi -lnMf) = upln(Mi/Mf) so…
 ∆v = upln(Mi/Mf)
Our Rockets
Engine specs: C6-5:
A8-3:
Total
Impulse
Time
Delay
Max Lift
weight
Max
thrust
Thrust
Duration
Initial
Weight
Propellant
weight
10.00
5 sec.
113g
3.4lbs
1.6 sec
25.8g
12.48g
Total
Impulse
Time
Delay
Max Lift
weight
Max
thrust
Thrust
Duration
Initial
Weight
Propellant
weight
2.5
3 sec.
85 g
2.4 lbs
0.5 sec
16.2 g
3.12 g
(A-series engine used only for test flights)
Liftoff!!!
Works Cited
•Canepa, Mark. Modern High-Power Rocketry. Baltimore, MD. Johns Hopkins University Press, 2003.
•Culp, Randy. "Rocket Equations." 25 March 2005. 25 May 2006.
<http://my.execpc.com/~culp/rockets/rckt_eqn.html>
•Hickam, Homer. Rocket Boys. New York: Random House. 1998.
•Nelson, Robert. "Rocket Thrust Equation and Launch Vehicles." June 1999. Applied Technology Institute. 25
May 2006. <http://www.aticourses.com/rocket_tutorial.htm>
•"Rocket Motion." 4 March 1994. University of Pennsylvania. 25 May 2006.
<http://www.physics.upenn.edu/courses/gladney/mathphys/subsubsection3_1_3_3.html>
•Sutton, George P. Rocket Propulsion Elements. Montreal: John Wiley and Sons. 2001.