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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10 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.

  10. Pingback: Q: If the Sun pulls things directly toward it, then why does everything move in circles around it? | Ask a Mathematician / Ask a Physicist

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