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The Mathematics of Rocket Propulsion

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.

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The Mathematics of Rocket Propulsion

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

  2. Rocketry in the Contemporary Age • Robert H. Goddard • Werner Von Braun and the V-2 Rocket • NASA • Military Applications • Amateur Rocketry

  3. Mathematical Relationships Critical to Understanding Rocket Propulsion • Impulse • Velocity • Acceleration

  4. 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.

  5. 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’

  6. Finding The Acceleration of a Rocket • Use Conservation of Momentum Pi=Pf • Pi=Mv , Pf= -dMUp + (dM+M)( v+dv); Where v is velocity of rocket, Up is velocity of propellant, and M is mass of rocket • Substitute Up=(v+dv)-up ; Where upis velocity of propellant 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)

  7. 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

  8. Finding the Velocity • Remember that-dMup= Mdv divide both sides by M • -dM/M up= dv integrate • ∫-up M-1dM = ∫dv; from Mi toMfand vito vf • -up (lnMf -lnMi) = vf -vi • up(lnMi -lnMf) = upln(Mi/Mf) so… • ∆v = upln(Mi/Mf)

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

  10. Liftoff!!!

  11. 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.

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