# Gravitational slingshots and energy conservation

I was watching YouTube video about gravitational slingshots and conservation of energy was explained as "planet loses its orbital energy around sun while spacecraft gains energy"

1. In solar system it explained as decreased orbital energy around sun but its supposed that spacecraft approach from front side (their initial velocities are opposite)

2. What if it approaches from back side, then due to conservation of momentum shouldn't final velocities of both increase? (same direction for initial velocities)

Actually when you think with reference to sun, both cases are ok, they don't violate anything, so it doesn't depend from what direction you approach planet with respect to its velocity. But what if there is no any solar/other systems and orbits and planet is stationary?

What i was thinking for is, lets consider big mass/planet in space but stationary this time, like really absolute stationary. When spacecraft gets near, due to centrifugal force it changes direction of its initial velocity lets say for matter of simplicity by 180 degrees (u shape turn). But due to conservation of momentum, planet accelerates in opposite direction little amount and thats where i got confused. Looks like total energy is increased.

I was thinking maybe it can be explained with the fact that nothing is stationary in space and there is no absolute speed as everything depends on which reference frame you are using, so there definitely should be some orbital path for that 'stationary planet' but then it violates my first assumption 'absolute stationary'.

When you starting do this thought experiment in completely stationary and empty space, things gets weird.

Consider a simpler example: an elastic collision of two bodies when there is a significant difference in masses.

By going into CoM reference frame, one can show that the velocity of a smaller body after the collision will increase by two velocities of a bigger body $$v_2=v_1+2u$$. What will happen if the larger body were flying away? It's as simple as letting $$u\to -u$$. The speed of a smaller body will decrease by two velocities of a bigger body.

In other words, instead of an accelerating slingshot effect, the spacecraft will experience decelerating slingshot effect. (And since the spacecraft lost it's kinetic energy, this energy was spent to increase the angular momentum of the planet, which counterintuitively resulted in an average slowing of the planet. However, it's another story)

It is convenient to view gravitational interactions as scattering. To illustrate how this works we take a look at the rest frame of the big planet and look only at the planet/spacecraft interaction for now. This gives the following picture

In this idealised scenario with only the planet and the spacecraft present we see that the spacecraft follows a hyperbola. This means at large distances the path asymptotically becomes a straight line with constant velocity (momentum). The difference between the initial momentum (right pointing blue arrow) and the final momentum (left pointing blue arrow) is the momentum kick $$\Delta p$$ that the spacecraft got from this flyby. Similarly the planet got a kick equal to $$-\Delta p$$ pointing in the other direction. But, as you mentioned, this will have neglible effect on the motion of the planet. If you think in terms of these asymptotic momenta and ignore the time interval where the planet and spaceship are interacting this problem is very similar to a hard sphere collision. The time interval where they are interacting is quite long so this comparison should be taken with a grain of salt.

Now by viewing this problem this way a lot of things should be more clear. In case there are no other planets this problem is the same as a hard sphere collision. So if you want to know why energy is conserved you could easily find a proof of that (for example in Vasily Mitch's answer). If you add back the sun you can reason in terms of this $$\Delta p$$ to find out if the energy will increase. Take the velocity vector of the spacecraft with respect to the sun. Is $$\Delta p$$ in the same direction as the velocity vector? The energy will increase and you will get a bigger orbit. Is $$\Delta p$$ in the opposite direction? The energy will decrease and you will get a smaller orbit.

• I see but it doesnt answer my question.Its helpful to approach it as collision problem but what i was trying to say,momentum-energy should be conserved.Gravitational forces act perpendicular to spacecrafts velocity.So it just changes it direction around 'peak' of that hyperbola.But same long term 'impulse' is acting to planet,which is eventually gains velocity little bit.So in order satisfy conservation of energy we see that final velocity vector of spacraft should lose some of its magnitude besides its directional change.But from where comes that loss if all forces acting perpendicular ? – Tymofei Ermirowich Gyuler May 11 at 10:37
• @TymofeiErmirowichGyuler "Gravitational forces act perpendicular to spacecrafts velocity" This is not true. This is only true at the point of closest approach. But for the two body system the asymptotic momenta still have the same magnitude. To see how the spacecraft gains energy you have to look at the complete system including the sun. If you are in orbit around sun and you can give the spacecraft a momentum kick in any direction you want, how would you orient that kick to increase the orbital energy around the sun? – AccidentalTaylorExpansion May 11 at 11:22
• I thought like this; on upper and lower areas of hyperbola you drawn,forces will cancel each other out(vertically) and same happens on peak.And you left with net horizontal force (deltaP) pointing to left. "But for the two body system the asymptotic momenta still have the same magnitude".Thats the point where i got confused.Beacause as planet is not moving,it cant lose its velocity/energy due to conservation but instead will increase in opposite direction. I would give momentum kick in direction of its initial velocity of coarse. – Tymofei Ermirowich Gyuler May 11 at 11:35
• But lets say you have stationary "explosive" near back of spacecraft and when it explodes,both of them will gain additional velocities (but opposite direction).Here i know that rise in kinetic energy is due to stored explosive energy,but i cant figure out reason for energy rise in first question where you have stationary planet. Maybe i should think like lost potential enery between spacecraft is analogical to electrical potential between to opposite charges that are infinitely far away from each other ? – Tymofei Ermirowich Gyuler May 11 at 11:38
• @TymofeiErmirowichGyuler You have to remember that both objects are orbiting the sun. In the rest frame of the planet the spaceship doesn't gain any energy. But in rest frame of the sun you do see a gain in energy. In the rest frame of the sun both the planet and spaceship are orbiting at very high velocities. If the momentum kick is in the same direction as this velocity the spacecraft will gain kinetic energy in the sun-frame (and the planet will lose some) – AccidentalTaylorExpansion May 11 at 12:11