1. Rocket Trajectories By Jan-Erik Rønningen Norwegian Rocket Technology [ contact@rocketconsult.no ]contact@rocketconsult.no [ www.rocketconsult.no ]www.rocketconsult.no Version: 1.50 2008

2. Contents  Different types of Rocket Trajectories  Typical Suborbital Trajectory  Guided Flight Trajectory  Homing Flight Trajectory  Circular Orbits  Gravity  Rocket Mass Ratio  Ideal Rocket Equation  Rocket Equation with Drag, Thrust and Gravity (2)  Terminal Velocity  Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (5)  Trajectory Simulation (2)

3. Different types of Rocket Trajectories  Free Ballistic Flight Trajectory – Gravity and Drag “Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket  Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits  Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring, ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a meeting point in time and space in order to hit the target  Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed change and gravity Satellites, space crafts, meteorites, planets etc.

4. Typical Suborbital Trajectory - Single-Stage Sounding Rocket  Flight Profile: Parabola (actually part of an ellipse)  Powered Phase: Rocket thrust with time  Coasting Phase: Free flight up to apogee controlled by gravity and drag mainly  Free Fall: From apogee to ground impact (or splash down)

5. Different types of Rocket Trajectories  Free Ballistic Flight Trajectory – Gravity and Drag “Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket  Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits  Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring, ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a meeting point in time and space in order to hit the target  Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed change and gravity Satellites, space crafts, meteorites, planets etc.

6. Guided Trajectory R0  h

7. Different types of Rocket Trajectories  Free Ballistic Flight Trajectory – Gravity and Drag “Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket  Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits  Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring, ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a meeting point in time and space in order to hit the target  Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed change and gravity Satellites, space crafts, meteorites, planets etc.

8. Homing Flight Trajectory X

9. Different types of Rocket Trajectories  Free Ballistic Flight Trajectory – Gravity and Drag “Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket  Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits  Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring, ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a meeting point in time and space in order to hit the target  Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed change and gravity Satellites, space crafts, meteorites, planets etc.

10. Circular Orbits R0 a = g G v

11. Gravity Earth is not a perfect sphere  g will therefore be a function of latitude  : g also decreases with the square of the distance, so for high altitudes a corrected g has to be calculated : R0=6371315m For h<120km R0 is so large in comparison to h that : [m] < 1% error

12. Rocket Mass Ratio Propellant mass (m d ) Rocket motor mass (m m ) Rocket inert mass (m r ) Payload mass (m p ) Burnout Mass Start Mass

13. Ideal Rocket Equation dm -ve M-dm v +

14. Rocket Equation with Drag, Thrust and Gravity (1)

15. Rocket Equation with Drag, Thrust and Gravity (2) Negative because mass is expelled in opposite direction of movement.

16. Terminal Velocity When drag equals gravity the net forces on the rocket is zero: Terminal Velocity is then: Higher velocity for objects that are heavier, more streamline, flying in lower atmosphere density and that has smaller frontal area.

17. Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (1) m r = rocket mass m m = motor mass m p = propellant mass g = const. = 9.81m/s 2 = const. = 1.22 kg/m 3 C D = const.

18. Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (2) Constant: Drag: t b = ? & v max = ? Simplify: Burnout time: Integration: t b = I t / T avg Normally t b is known. NOTE:

19. Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (3) Finding v max :

20. Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (4) We now want to find the burnout altitude, h b :

21. Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (5) We now want to find the ascent altitude, h a and H: H = h Coast + h b

22. Trajectory Simulation (1) Input values Results ”Launch” Software Download: http://users.cybercity.dk/~dko7904/software.htm

23. Trajectory Simulation (2) Burnout Apogee