Space ships can use planets to move.
Space ships can fly near big planets to move. 
Spacecraft can use planets to change their path. This is called a gravity assist.
A spacecraft flies near a large planet. The planet's gravity pulls on the ship. The ship gains speed from the planet's motion. This happens as the planet orbits the Sun. The ship can also slow down. This happens if the ship passes in front of the planet. 
When a ship gains speed, the planet loses a tiny bit. This follows Newton's Third Law. The planet is so big that its speed does not change much. A tiny ship cannot slow down a huge planet like Jupiter.
Scientists use this to plan long trips. In the 1960s, Gary Flandro found a rare alignment. The outer planets were in the right spots. This helped the Voyager probes visit many planets.
A gravity assist is a clever way to move through space.
How does this work? A spacecraft flies close to a large planet. The planet's gravity pulls on the ship as it passes by. The ship actually takes some energy from the planet's own motion around the Sun. 
People have studied these paths for a long time. Yuri Kondratyuk suggested using moons to speed up ships in a paper from 1918. Friedrich Zander also understood this physics in 1925. Later, Gaetano Crocco calculated trips using many gravity assists in 1956. The first time it was actually used was in 1959. The Soviet probe Luna 3 used it to photograph the far side of the Moon. 
There are important numbers behind these big moves. A typical space probe might weigh only one metric ton. Jupiter is much larger, with a mass of almost 2 x 10^24 metric tons. Because Jupiter is so huge, the probe's pull does not really change the planet's speed.
You can think of this like a tennis ball hitting a moving train.
A gravity assist is a specialized spaceflight maneuver used to change a spacecraft's path or speed. 
The mechanism of a gravity assist relies on the relative movement of a planet. As a spacecraft enters the gravitational sphere of influence of a planet, it is pulled by that planet's gravity. The assist is provided by the motion of the gravitating body itself. If the spacecraft's trajectory is designed to pass behind the planet, it will gain speed. If the trajectory passes in front of the planet, the maneuver acts as a braking maneuver to decrease speed.
To understand the physics, consider the frame of reference. In the planet's own frame of reference, the probe leaves with the same speed it had when it arrived. However, when viewed from the Solar System's frame of reference, the speed change is very apparent. A helpful analogy is a tennis ball bouncing off a moving train. If you throw a ball at a train approaching at 50 km/h, the ball's departure speed relative to the platform is much higher. The ball adds the train's velocity to its own. In space, the planet acts like that moving train, providing a massive boost to the smaller probe.
This exchange of energy follows Newton's Third Law. Any gain in kinetic energy by the spacecraft results in a corresponding loss for the gravitational body. The linear momentum gained by the spacecraft is equal in magnitude to the momentum lost by the planet. However, the mass of a planet is many orders of magnitude larger than a spacecraft. For example, a typical probe might weigh one metric ton. Jupiter has a mass of almost 2 x 10^24 metric tons.
The history of this concept involves many different researchers. Yuri Kondratyuk suggested using moons to accelerate spacecraft in a paper dated 1918–1919. Friedrich Zander also demonstrated a deep understanding of this physics in 1925. In 1956, Gaetano Crocco calculated interplanetary journeys using multiple gravity assists. The first actual use of the maneuver occurred in 1959. The Soviet probe Luna 3 used a gravity assist to photograph the far side of the Moon. Later, Michael Minovitch developed techniques at NASA's Jet Propulsion Laboratory that led to the famous Planetary Grand Tour.
One of the most significant applications was the discovery of a rare planetary alignment. In 1964, Gary Flandro studied the outer planets and found a specific alignment of Jupiter, Saturn, Uranus, and Neptune.
There are physical limits to how these maneuvers are used. The main limit is that planets are rarely in the correct positions for a specific destination. A similar alignment of the outer planets will not occur again until the middle of the 22nd century. Another limit is the distance of closest approach. The magnitude of the velocity change depends on how close the probe gets to the planet. The atmosphere of a planet can also limit how close a spacecraft can safely fly. If more speed is needed than gravity provides, engineers use a rocket burn at the periapsis. This is known as the Oberth effect, where burning fuel at the highest velocity provides the most kinetic energy.
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