OT: Physics Questions

[Gavin786]

I am planning on making some mods for KSP so thought I need to brush up on my high school physics and something is annoying me and maybe some Kerbonauts more knowledgeable than me in this area can help.

So here is what is annoying me.

There is something called the Oberth Effect which means that at the Periapsis of an Orbit(when the space ship is going faster), changes to velocity impart more kinetic energy than they would do at the Apoapsis when the ship is going slower.

This is because Ek = 0.5mv^2

Now the Force applied at either periapsis or apoapsis is the same as F=ma, m is the same and a is the same(our delta v does not change based upon ship speed), so the same force for same amount of time is applied at both points in the orbit yet a different amount of kinetic energy is imparted to the space ship?!? 

Isnt there something called the law of conservation of energy?  Energy imparted is same in both cases but gives different amounts of kinetic energy?  What am I missing ?

Any help with that would be appreciated.

Gavin786

[Vanamonde]

Since there is actual space science behind your question, it has been moved to Science & Spaceflight, where the math guys hang out. 

Personally, I believe the Oberth effect is black magic and can be summarized as, "If you go faster, you go faster." But somebody who actually understands it will be along presently to give an explanation. (Which I will neither understand nor believe.) 

[Snark]

1 hour ago, Gavin786 said:

Energy imparted is same in both cases

Careful with your terms.

The impulse (force times time) is the same in both places.  Energy isn't.  Impulse != energy.  Don't mix up momentum and kinetic energy, they're not the same thing.  One goes linearly with the velocity and is a vector; the other goes with the square of velocity and is a scalar.

Energy is force times distance, and the faster-moving ship is applying that force over a greater distance.

[sh1pman]

It's actually easily explained by a combination of the law of conservation of impulse and the law of conservation of energy. Conservation of impulse means that the impulse gained by a ship equals to the impulse imparted to exhaust gases by its rocket engine. If the exhaust velocity is the same (it usually is) and the burn time is the same, then the impulse gained by ships at periapsis and apoapsis is also the same, resulting in the same velocity gain.

Next, conservation of energy. The change in kinetic energy of the exhaust must be equal to the change in ship's kinetic energy. Remember that kinetic energy of stored propellant is not zero, it scales with the second power of ship's orbital velocity. Kinetic energy lost by the propellant equals to

dE = 0.5*M*(V1^2 - V2^2)

Where M - mass of the burned propellant, V1 and V2 - its orbital velocities before and after the burn. Let's say that V2 = V1 - dV. Then,

dE = 0.5*M*(V1^2 - (V1 - dV)^2) = 0.5*M*(V1^2 - (V1^2 - 2*V1*dV + dV^2)) = 0.5*M*(2*V1*dV - dV^2).

As you can see, kinetic energy lost by propellant scales with V1, orbital velocity at the start of the maneuver. The faster you go, the more kinetic energy loses your propellant when it's burned by your rocket engine. Conservation of energy means that your ship gets all kinetic energy lost by the propellant. Which, in turn, means that the faster you go, the more kinetic energy you can extract from your propellant. That's Oberth effect. 

[sevenperforce]

It's magic. Magic math goblins come behind your ship and push.

Spoiler

In all seriousness, though...

The Oberth effect works because of gravity. When a spacecraft swings around a planet, gravity pulls it in, then slingshots it back out. It's easy to think of a hyperbolic trajectory as nothing more than a curved path through space, but it's not. The fabric of space, bent by the planet's gravitational mass, acts like an invisible spring, stretching as you zip around it and then relaxing as you careen away. Basic physics teaches that a compressed spring holds potential energy; the same thing is happening (in reverse) during a hyperbolic flyby.

Space is stretchy, and that stretchiness allows us to do some cool things. On a trampoline, you can bounce higher if someone else has just jumped before you, because the surface is already stretched. In the same way, if you burn a small amount of propellant before you approach near another planet, you only get a small amount of dV. You'll hit the gravity well, stretch it, and then be slung out. But if you wait until your momentum has already stretched the fabric of space, and burn the same amount of propellant, then your thrust has something extra to push against and so you emerge going much faster.

You can also conceptualize it using the forces acting on the spacecraft. Why do you speed up when you are falling toward a planet, then slow down as you are slung away? Well, gravity is pulling you in as you approach, and then it's trying to pull you back as you escape. It uses the same amount of force pulling you in that it uses to try and pull you back. However, when you burn your engine at closest approach, gravity is no longer just pulling on your ship; part of that total force is now being exerted on the propellant you just dumped. You're now moving faster, meaning that gravity has less time to pull you back, and the propellant is now moving slower (relative to its original speed), meaning gravity has more time to pull on it. So total energy is conserved, but you get a boost because gravity spent more time tugging you in than it did pulling you on your way out.

[Cunjo Carl]

Plenty of good answers already, but I'll add a bit. We actually have an equivalent to the Oberth effect right here on Earth, if you've ever ridden a bike in a hilly area. When faced with a short hill, people will speed up extra at the bottom rather than trying to fight the hill extra at the top.

Especially on a bike, our legs act a bit like rocket engines. Our legs can provide a force for a certain time before getting tired, just like rocket engines can provide a force for a certain time before running out of fuel. In either case, that force per time translates into a certain amount of 'speed up'. In other words, if you crank the pedals harder for a little bit, you'll be going faster. Great!

So if you're toodling along at a normal speed (say 16kph or 10mph) and you notice the road in front of you has a short but steep incline (say 2m or 6.5ft tall) you'll naturally speed up a bit so that you hit the incline with plenty of speed and whisk over it, rather than needing to spend the extra energy by slowing to a crawl halfway up and cranking your way up the second half. Let's compare how much speed (aka. energy) we come out with if we speed up another 6.5kph (4mph) at the bottom of the hill while we're still going fast rather than trying to use it at the top of the hill after it's too late and we've already slowed down.

1. Use the 6.5kph speedup first to get to 22.5kph and hit the hill: you'll come out traveling 16kph just like originally! This is what people normally do because it's the easy way to ride a bike up a hill.
2. Hit the hill going 16kph, and naturally slow down to 1.5kph by the top. Then use the 6.5kph speedup and you'll only be going 8kph- half your original speed! People don't do this. It's hard!

Anyways, the Oberth Effect is all the same things for the same reasons, except IN SPAAAACE .  Which is why it seems more magical

 

[Bill Phil]

In escape trajectories the Oberth effect appears in an interesting way.

Hyperbolic excess velocity (the velocity relative to the gravitating body at an infinite distance) can be found using this equation:

v(hyperbolic excess) = sqrt(V^2 - v(escape)^2)

Since characteristic energy (C3) is hyperbolic excess velocity squared:

C3 = V^2 - v(escape)^2

V is the spacecraft’s velocity and v(escape) is the escape velocity at the spacecraft’s current distance from the gravitating body. 

When the escape velocity is high (such as in LEO) the total velocity required to reach a C3 of, say... 14.3 (InSight’s C3) is only 637 m/s past escape velocity. 

When escape velocity is low (such as in GEO) the total velocity required to reach C3 = 14.3 is 1414 m/s faster than escape velocity. And of course you had to spend the energy to get to GEO... 

But this shows that a Mars trajectory doesn’t take too much delta-v to get to - after you’ve reached escape velocity.

[Ol’ Musky Boi]

I'm not sure how accurate this explanation is, but it's how I've always understood it.

If you, for example, wanted to go from an elliptical orbit to an escape trajectory, what you really want to do is speed up to escape velocity. You could apply this change in velocity at apoapsis, where your orbital speed is slowest, but that would mean you have to speed up a lot more to get to escape velocity. If you apply that change in velocity at periapsis, where you are already moving very fast, then you have to speed up less than you would have to at apoapsis. It doesn't violate conservation of energy, because at apoapsis your kinetic energy becomes stored as gravitational potential energy, and at periapsis your gravitational potential energy becomes kinetic energy (i.e more speed). The Oberth effect is all about making use of all that pent up gravitational potential energy rather than letting it go to waste.