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Rocket Propulsion, Classical Relativity, and the Oberth Effect Philip Rodriguez Blanco, Grossmont College, El Cajon, CA Carl E. Mungan, U.S. Naval Academy, Annapolis, MD

THE PHYSICS TEACHER ◆ Vol. 57, October 2019 441 Supplement to “Rocket propulsion, classical relativity, and the Oberth Effect”, published in The Physics Teacher, 2019 Authors: Philip Blanco, Grossmont College Carl Mungan, United States Naval Academy Contents: Figure 1 (without annotations) Alternate Figure 1 Additional figures Animations with Systems Tool kit (separate PPT File): Impulse fired at periapsis (cf. Figure 3) Impulse fired at apoapsis for comparison Figure 1 (annotations removed) This rocket-powered skateboard is about to roll downhill. To escape on the right with the maximum speed, where should it fire a brief impulse? (Or does it not matter?)

A C

B

Thanks to student Devin Potratz for assistance with the graphics Alternate Figure 1

Thanks to student Devin Potratz for assistance with the graphics

This rocket-powered skateboard is about to roll downhill. To escape on the right with the maximum speed, where should it fire a brief impulse? (Or does it not matter?)

A C

B You get moreΔK for a fixed Δv if you are already moving fast! 100 K = ½ Explanation: mv2 80 Impulse F δt = mΔv is fixed, but Work = F δx =F vδt ΔK increases with speed. 60 Δv

40

But where does this “extra” 20 kinetic energy come from? ΔK Kinetic Energy per kg (J/kg) Δv 2 4 6 8 10 Speed (m/s) Where does this “extra” kinetic energy boost come from?

v Δv ex M mex R v v For the rocket: “extra” 2 ΔK = 1 M v + Δv − 1 M v2 = 1 M Δv2 + vM Δv R 2 R ( ) 2 R 2 R R For the exhaust: 2 ΔK = 1 m v − v − 1 m v2 = 1 m v2 − vm v ex 2 ex ( ex ) 2 ex 2 ex ex ex ex Since mexvex = MRΔv, when we add up the total ΔK, ⇒ ΔK = ΔK + ΔK = 1 M Δv2 + 1 m v2 = ΔE total R ex 2 R 2 ex ex chem - invariant! Classical Relativity è invariance of impulse and total energy change (Rocket converts same chemical è kinetic energy independent of speed.)

For a rocket already moving fast, in our reference frame, • more ΔKtotal goes to the rocket…and less to the exhaust, so • rocket “steals” kinetic energy from exhaust! Orbital Oberth effect (Figure 3)

2GM Planet surface escape speed v = P SE R P Set rocket mass MR ≡16 mex 2 →ΔEchem = 17/32 mex vex Set exhaust velocity vex ≡ 1.46286 vSE 10 →Δv = vex/16 = 0.09143vSE Animation at →ΔE = 2.274 GMm /R . chem ex P https://vimeo.com/286053315 Initial rocket+fuel orbit (ellipse):

Set semi-major axis a ≡ 7.50 RP, eccentricity e ≡ 0.8.

→periapsis distance = 1.5 RP, →periapsis speed = 0.77460 vSE 5 17 → Etotal,i = - /15GMmex/RP = -0.4984 ΔEchem

Resulting rocket orbit (hyperbola):

Semi-major axis a = -6.00 RP, eccentricity e = 1.250. 4 → Etotal,f = + /3GMmex/RP →ΔER = +2.4 GMmex/RP =+1.0555 ΔEchem .

Resulting exhaust orbit-10 (ellipse): -5 5 10

Semi-major axis a = 2.591 RP, eccentricity e = 0.4211. →ΔEex = -0.1263GMmex/RP = -0.0555 ΔEchem. B

-5

-10