Kdjykuj

I was reading about gravity assist and I read that both the voyagers used gravitational force of these planets to speed up.

But I wanted to know why weren't they completely attracted by their gravitational field?

One thing I thought about it was that the escape velocity of these planets were known to NASA and these space crafts used the propeller fuel to escape but how did NASA calculated escape velocity?

$$v_{escape} = sqrt{frac{2GM}{R}}$$

But how could they predict mass?

Radius I guess can be predicted by parallax method.

why weren't they completely attracted by their gravitational field?

How much a trajectory is changed, depends on 3 factors:

1. the mass of the planet,


2. the speed of the spacecraft,


3. the distance between spacecraft and planet

Voyager's speed and distance were chosen to make sure Voyager wouldn't enter orbit around the planet. Voyager's speed before approaching Jupiter was higher than Jupiter's escape speed.

In fact, as long as your trajectory doesn't touch the planet's atmosphere, you're good: because you're arriving from another planet your speed is always higher than escape speed.

But how could they predict mass?

When a planet has moons, you can calculate the planet's mass to a good degree of accuracy. This uses Kepler's third law:

$$ M = frac{4pi^2r^3}{GT^2}$$

G is the universal gravitational constant, 6.6726 x 10-11N-m² /kg²

r = orbital radius of the moon

T = orbital period of the moon

these space crafts used the propeller fuel to escape

No. When Voyager 1 was launched from Earth, the last stage of the rocket put it on an escape trajectory, then it was jettisonned. That was the last time a rocket was used to significantly change Voyager's speed.
It traveled to Jupiter without further propulsion, so its speed slowly decreased as it got further away from the Sun.

Here's a plot of Voyager 2's speed:

Then as it got close to Jupiter, Voyager's speed increased drastically as it was pulled in by Jupiter's gravity. At Voyager's closest approach, Voyager's trajectory was bent. This put the spacecraft on course to Saturn.
As Voyager sped away from Jupiter, Jupiter's gravity slowed it down.

Let's try and understand how gravity works in space. This is kind of key idea to understanding lots of issues in space travel and astronomy.

So imagine a space probe, or rock, which is heading in from deep space, aimed almost, but not quite towards a planet. We can break that motion into two parts -- the part towards the planet and the "sideways" part. The planet's gravity accelerates the part towards the planet, but it does nothing to the sideways part, so the space probe is still not heading quite towards the planet. This keeps on happening, so the probe is moving increasingly fast, still almost but not quite heading at the planet.

After a while the probe whizzes past the side of the planet, going very fast. It can't hit it because it has never lost the "sideways" part of its motion (called angular momentum, more or less). It is still being pulled towards the planet but still moving around it, due to all the speed it has picked up. Then that speed starts to carry it away from the planet again (possibly with its course having been bent around to a greater or lesser extent). Now the gravity is slowing it down. However it is slowing it down just exactly as much as it speeded it up coming in so if it came in from far away at a certain speed, it will eventually escape equally far away at the same speed.

If someone can add some diagrams to this, that would be great.

To add to the answers @Hobbes & @Steve Linton posted, the mission designers indeed knew Jupiter's gravity field quite well from the orbits of Jupiter's moons. But before the Voyagers arrived they got additional measurements from the close flybys of two other spacecraft, Pioneers 10 and 11.

@Steve Linton correctly describes the effect of the "sideways" part of a spacecraft's velocity with respect to the planet it's approaching. If the sideways component is large enough, the probe will miss the planet. To increase that sideways component, on approach you aim the spacecraft [via trajectory adjustment maneuvers, where you use the spacecraft's rocket engine(s)] to precisely tune the components of velocity, one straight toward the planet and one sideways. That combination sets the "aim point", the point to the side of the planet where the spacecraft would go if the planet had no gravity. Trajectory designers call that miss distance "b", a part of the b-plane aim point.

This diagram helps to see the effect of increasing b if the approach velocity is held constant. For typical interplanetary trajectories the approach velocity doesn't depend on the chosen b. [Sorry for the pixellation—this is a very old diagram I pulled out of my presentation archives]

The brownish-tan (in the original file it was orange!) circle represents the planet, and the different colors of curves represent the trajectories followed by spacecraft with different b, all coming in at the same approach velocity and parallel to the black dashed line through the center of the planet. The thinner, dot-dashed lines represent what the spacecrafts' trajectories would be if the planet had no gravity. If b isn't large enough, as with the red and brownish-tan (formerly orange!) trajectories, they impact the planet. The thick, dashed red and brownish-tan lines show how those two trajectories would continue if they didn't run into anything, as would be the case if the planet were the same mass but a lot smaller in size (i.e., a lot denser).

The green and blue lines are for trajectories whose b are large enough that they miss the planet. They point out a couple of characteristics of these hyperbolic orbits: 1) the larger the b, the farther from the planet is the closest-approach distance; and 2) the larger the b, the smaller is the angle the planet "bends" the trajectory. Again, this is for a fixed approach velocity. The red trajectory is bent by almost 180°, while the blue one is bent only ~135-140°.

This bend is key to making gravity assists so useful.

Pretty much the speed of the aircraft was really high, so the gravitational force around it couldn't bring it crashing down. But most likely the spacecraft changed it's path a little due to the slight force.