How the gravitational slingshot works

The gravitational slingshot is something I'd heard about but had never bothered to look up - until now. It sounded like magic: you fire a spaceship towards a planet and (then a miracle occurs before) it comes out moving faster. It's how the Voyager spacecraft picked up enough speed to leave the solar system, and - now I've read up on it - it seems like genius.

Before we get into interplanetary pinball, let's think first about the more conventional, Earth-bound version. If the ball hits your flipper, it just bounces unimpressively off of it. If you have a perfectly smooth and elastic1 collision, the ball bounces off with the same speed as it arrived - only reflecting one of the components so the direction changes.

If the flipper's moving, though, it's a different story. If you were sitting on the flipper2, you'd see exactly the same thing in both situations: the ball comes towards you at some speed, it bounces away with the same speed.

But, if you're not on the flipper, you see something altogether different: because the flipper is moving, the approach speed of the ball to the flipper is faster than the speed of the ball. That means the separation speed is also higher. In fact, it's higher still: you have to take into account that the ball is separating from something moving in the same direction.

If the ball is travelling with speed $u$ towards a flipper which has speed $v$ (and they're moving towards each other), the approach speed is $u+v$. The separation speed is also $u+v$, and that's the difference between the ball's speed $U$ and the flipper's speed $v$3. So, $U-v = u+v$, which gives $U = u + 2v$ - the flipper has given the ball a big boost in speed.

(Aside: I know I should be doing this with velocities; I find the speed version easier to get hold of.)

So, what does all this have to do with spaceships? Funny you should ask: it's almost exactly the same idea. You fire a small object - a spacecraft - at a much bigger moving one - a planet or a moon - and you get a boost in speed. The only difference is, that instead of crashing into the planet (which would be a definite tilt), you use its gravity to change your direction.

It's not easy (for me) to explain why the gravitational effects of the planet work the same way as a pinball flipper - effectively, the planet applies a force for some time, which (dimensional analysis ahoy!) gives an impulse on the spaceship - just like the flipper applies to the pinball. Because the planet is moving, the approach and separation velocities go through the same kind of trickery as before - the spaceship doesn't just carry on with the same speed, it's affected by the planet.

But what about conservation of momentum? Aren't we getting something for nothing here? Well... sort of. If you followed the footnote, you'll have noticed that I mentioned the flipper was much more massive than the ball. This is why trains don't slow down when they run into bees. Or rather, they do, but it's by such a small amount you probably wouldn't spot it. Because a planet is much bigger than a spacecraft, an impulse changes its speed less. With numbers: Voyager I had a launch mass of 700kg. Jupiter has a mass of 1,900,000,000,000,000,000,000,000,000kg, give or take. If Voyager I had used the slingshot to accelerate to the speed of light (which it couldn't, obviously), it would have slowed Jupiter down by $\frac{mv}{M}\simeq \frac{700\times 3 \times 10^8}{2\times 10^{27}} \simeq 10^{-18}$m/s - or about three centimetres per billion years. Jupiter wouldn't notice.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. meaning no energy is lost in the collision []
  2. hold on tightly! []
  3. assuming the flipper is much more massive than the ball []

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2 comments on “How the gravitational slingshot works

  • Cav

    Superb post! This slingshot effect is something I’ve know about for years but never really looked into. I think I may have to investigate it further!

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