The Mathematics of Rocket Propulsion  Ben Ferguson  Abhishek Gupta  Matt Kwan  Joel Miller BY:

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 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’: 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 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 Use Conservation of Momentum P i =P f P i =Mv, P f = -dMU p + (dM+M)( v+dv); Where v is velocity of rocket, U p is velocity of propellant, and M is mass of rocket Substitute U p =(v+dv)-u p ; Where u p is velocity of propellant relative to the rocket Mv= -dM(v+dv-u p ) + (dM+M)(v+dv) then use the distributive property Mv= -dM(-u p ) -dM(v+dv) + dM(v+dv) + M(v+dv)

Finding The Acceleration of a Rocket Mv= -dM(-u p ) -dM(v+dv) + dM(v+dv) + M(v+dv) Mv= -dM(-u p ) + M(v+dv) Mv= dMu p + Mv + Mdv 0= dMu p + Mdv -dMu p = Mdv divide both sides by dt -dM/dt u p =Mdv/dt -dM/dt is rate of fuel consumption and dv/dt is acceleration a -dM/dt u p is known as thrust T so… T=Ma

Finding the Velocity Remember that -dMu p = Mdv divide both sides by M -dM/M u p = dv integrate ∫-u p M -1 dM = ∫dv; from M i to M f and v i to v f -u p (lnM f -lnM i ) = v f -v i u p (lnM i -lnM f ) = u p ln(M i /M f ) so… ∆v = u p ln(M i /M f )

Our Rockets Engine specs: C6-5: A8-3: (A-series engine used only for test flights)

Liftoff!!!

Works Cited Canepa, Mark. Modern High-Power Rocketry. Baltimore, MD. Johns Hopkins University Press, Culp, Randy. "Rocket Equations." 25 March May Hickam, Homer. Rocket Boys. New York: Random House Nelson, Robert. "Rocket Thrust Equation and Launch Vehicles." June Applied Technology Institute. 25 May "Rocket Motion." 4 March University of Pennsylvania. 25 May Sutton, George P. Rocket Propulsion Elements. Montreal: John Wiley and Sons