Q: How does a gravitational sling shot actually speed things up?

Physicist: A gravitational slingshot (or “gravity assist”) is a slick way to pick up speed using a moving planet’s gravity.  What’s confusing about the gravitational slingshot is that, from the point of view of the planet, the object in question comes flying in from space (\vec{U}) with some amount of kinetic energy, and leaves (\vec{W}) with the same amount of kinetic energy (conservation of energy).  So how can it speed up?

Here’s the slickness: the Galilean Equivalence Principle.  The GEP states that the laws of physics work the same whether you’re moving (at a constant speed) or not.  So if you look at the exact same situation from another perspective, where the planet is moving, you’ll notice that the incoming and outgoing speeds are different.

An object that passes by a stationary planet will approach and leave at the same speed (but in different directions of course). However, if the planet is moving, then the incoming and outgoing speeds are different.

Here’s voyager 1 and 2 doing their slingshot thing.

The total change in velocity, \Delta, experienced by the slungshot object is |\Delta|=2|\vec{U}|\cos{\left(\frac{\theta}{2}\right)}, where |\vec{U}| is the incoming speed (from the planet’s perspective), and \theta is the angle between the incoming and outgoing trajectories (again from the planet’s perspective).  So in general, a sharper angle yields a bigger boost.

The course of the Galileo probe. After it's initial launch it did three slingshots, around Venus, Earth, and Earth again, to gain enough speed to get to Jupiter's orbit.

A slingshot increases the kinetic energy of the object in question by decreasing the kinetic energy of the planet.  But don’t worry too much, an ant pushing a tricycle is having about one hundred quadrillion (1017) times more effect.  Gravitational slingshots are used primarily for the probes we’ve sent to the outer solar system.  It lowers the fuel costs a lot, but all the wandering around makes the trip quite a bit longer.

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23 Responses to Q: How does a gravitational sling shot actually speed things up?

  1. Pingback: Q: Can we build a planet? « Ask a Mathematician / Ask a Physicist

  2. The Cool Dude says:

    If I had a profound distaste for Mars, could I theoretically slingshot a bunch of big rocks around it until it fell into the sun? How long would this take? About how much rock material would be needed?

  3. The Physicist The Physicist says:

    You could. It would require on the order of a Mars’ worth of material to do.
    If you were in a hurry I suppose you could do it all at once, with a single object (if you can find one big enough).

  4. James says:

    What’s the numerical figure for a gravitational slingshot past Jupiter? Thanks.

  5. Elaine J says:

    I’m writing a story in which humans from the future have learned how to use the ‘slingshot effect’ to travel back in time. Is this theoretically possible and how can I have my characters explain it so that it makes sense without being too long-winded and technical – regardless of whether the theory holds any water? (This is fiction, after all) – any response would be gratefully received (if if it is just to say that it is all bunkum!)…

  6. Mike D. says:

    Why doesn’t the earth decelerate the object back to its incoming velocity as it heads away from the earth?

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  8. Kurt S. says:

    Or, if you like Mars on the other hand you could hurl comets at it in a trajectory that decelerates it when they get captured by his grav field at the same time dropping some oceans on him and putting him on a (slightly) smaller orbit around the sun… You’d have to find just the right comets though… i doubt this would be feasible on a terraforming time table of few centuries.

    btw. very nice explanation, always wondered where the deltaV would come from.

  9. Jim says:

    @Elaine: If you’re talking Star Trek 4 type of slingshot, that was based on an episode in TOS, so you could look that up. However, in reality, the Sun’s mass is far too small to distort space-time sufficiently. What you need is a black hole that rotates rapidly. A pulsar (neutron star) might do too. Rotating massive objects twist space-time, and it can “pile up”. The ride would be rough as you fly around it very close to the event horizon, but there are enough unknowns and exotic things going on that you could easily contrive just about any plausible fiction you wanted.

    Or you could try to rewrite the story such that the mechanism is irrelevant or unknown; many time travel stories do this.

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  11. Sofa says:

    Is the decrease in kinetic energy (rotation?) of the planet perminant? Could the planets rotation be increased by an identical but reversed approach?

  12. Mihir H says:

    Why doesn’t the earth decelerate the object as it heads away from the earth?
    And how is the object launched out of it’s orbit around the earth once it achieves the needed velocity ? Please reply asap

  13. The Physicist The Physicist says:

    @Mihir H
    The Earth does decelerate the object as it moves away, but not as much as it accelerates it overall. Objects that are slungshot are never in orbits, they’re merely passing by. If an object is in orbit, then it’s stuck there.

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  16. jens-erik says:

    Suppose human were to travel in a spacecraft that accelerates by gravity assist – what would the g-force on the body be?

  17. J. Rich says:

    That doesn’t make a shred of sense. The same gravitational force that helps you by being “pulled” on the backside of the planet ( so as to go along with it as well as adding your own velocity) is the same gravitational force that is suddenly going to let you go when exiting at this precise time, in the same direction the planet is now moving. Newsflash: you can’t even find missing planes that take off out of Indonesia.

  18. Zak says:

    What is the max velocity gained by a sling shot (say from the moon)? Maybe using a spaceship traveling at 100k mph to start?

  19. kenneth hunt says:

    I am at present an old man—but when I was in the navy ( WWII) in the radio lab we
    had a large —very large radar magnet—-it had a keeper in it about the size of a hockey puck—–we used to take the WWII pennies and try and hit the “sweet spot” and have them “sling shot” around the magnet—–one out of ten trys would work.
    on the tenth you would get hit in the head—-by this time one has dropped their guard.. I would like to build this toy—-with the two —100 pound pull magnets
    that I have—would you make some suggestions please

  20. Thomas Coates says:

    “To reiterate: the correct theory of gravity – that we know – General Relativity – says that gravity is the curvature of spacetime due to matter and energy. In curved spacetime, masses follow geodesics – straight paths. That is – they do not accelerate. And because they do not accelerate no force is needed to change their velocity (Newton’s Second Law is still universally true, even given General Relativity. Observations falsify Newton’s Law of Gravity: but not his laws of motion (see note [1] at the end of this article). Because in 4 dimensional spacetime they are not changing their velocity. They are following the shortest path between points in curved space around a mass.”

    Based on this… how is it possible that gravity can accelerate an object over and over again if its already traveling at ‘terminal velocity’ from the first slingshot maneuver and its not losing any energy traveling through space?

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  23. Eric Thorpe says:

    Space-time theory does not disprove the law of gravity. We don’t possess the means to determine whether or not the relative time-shifting one experiences at high velocity is caused by gravity or some other, unknown phenomenon. We know that they are somehow related, but everything else is guesswork. It does, however, provide a convenient way to illustrate the law of gravity to someone who has trouble with spatial reasoning.

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