The Mathematics of Rocket Propulsion BY: ØBen Ferguson ØAbhishek Gupta ØMatt Kwan ØJoel Miller
Rocketry in the Contemporary Age è Robert H. Goddard è Werner Von Braun and the V-2 Rocket è NASA è Military Applications è Amateur Rocketry
Mathematical Relationships Critical to Understanding Rocket Propulsion Impulse l Velocity l Acceleration l
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 l Pi=Mv , Pf= -d. MUp + (d. M+M)( v+dv); Where v l Use Conservation of Momentum is velocity of rocket, Up is velocity of propellant, and M is mass of rocket l U =(v+dv)-up ; Where up is velocity of propellant Substitute p relative to the rocket l Mv= -d. M(v+dv-up) + (d. M+M)(v+dv) then use the distributive property l Mv= -d. M(-up) -d. M(v+dv) + M(v+dv)
Finding The Acceleration of a Rocket l Mv= -d. M(-up) -d. M(v+dv) + M(v+dv) l Mv= -d. M(-up) + M(v+dv) l Mv= d. Mup + Mv + Mdv l 0= d. Mup + Mdv l -d. Mup= Mdv divide both sides by dt l -d. M/dt up =Mdv/dt l -d. M/dt is rate of fuel consumption and dv/dt is acceleration a l -d. M/dt up is known as thrust T so… l T=Ma
Finding the Velocity -d. Mup= Mdv l -d. M/M up= dv integrate l Remember that divide both sides by M l ∫-up M-1 d. M = ∫dv; from Mi to Mf and vi to vf l -up (ln. Mf -ln. Mi) = vf -vi l up(ln. Mi -ln. Mf) = upln(Mi/Mf) so… l ∆v = upln(Mi/Mf)
Our Rockets Engine specs: C 6 -5: A 8 -3: (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/subsubsection 3_1_3_3. html> • Sutton, George P. Rocket Propulsion Elements. Montreal: John Wiley and Sons. 2001.
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