For example:
Is there some intuitive way to explain what angle a probe needs to enter and leave a body's gravity well to perform a gravity assist to speed-up or slow-down?
Let's start by understanding the mechanism of a gravity assist. As a spacecraft approaches a planetary body, it gets affected by the planets gravitational pull. Getting nearer, the pull increases, and eventually when the spacecraft passes the planet, the pull decreases.
If you think about a stationary planet as an absolute frame of reference, picture throwing a body near it. The body will curve towards it, accelerating, then pass it, and then carry on, eventually retaining its original speed.
However as you know, planets are not stationary but rather, they orbit around the Sun at speeds to the order of tens of kilometres per second. Hence any spacecraft affected by the planets gravity, is affected by its orbital speed too.
Let's use a simple analogy.
Imagine a girl on a merry-go-round. She is sitting on the edge, with her hand stretched out, and the the merry-go-round is rotating at a certain speed counter-clockwise. You throw a ball at her hand. She catches it for but a second, and then lets go. The ball is observed to accelerate during this process.
In this analogy, the girl was a planet, the merry-go-round was her orbit, and her hand was the gravitational pull of that 'planet'. This is highly simplified explanation that serves only to explain the directions involved. If you throw a ball from behind her as she moves away from you, the ball will accelerate.
It's very easy to now imagine the inverse of this analogy; what if you throw the ball to the girl head on, as she comes closer to you. You can intuitively tell that the ball will decelerate during this process.
There are many other analogies, like a ping-pong ball hitting a ceiling fan, or a skateboarder grabbing a car on a round-about. The spacecraft is basically stealing the angular momentum of the planet.
Once you have these basics down, the rest of gravitational mechanics are easy to comprehend. The following diagram will show you the change in direction observed in an accelerating assist.
A more complex representation of what a gravity assist looks like when the planet is approached from behind. (Wikimedia)
For a real-world example, take Cassini-Huygens (my favorite spacecraft). Cassini's mission plan looked like this:
Orbital diagram of the path Cassini took to eventually reach Saturn (Wikimedia)
Cassini made a total of four flybys, all to gain speed. From what we've learnt earlier, acceleration occurs when we approach a planet from behind. Take a closer look at the diagram above and you can see that all the assists approached their respective planets from behind them in their orbits. This is the fundamental principle of gravity assists, observed in the real world.
Cassini's speed plotted over time, with the gravity assists labelled. Using multiple assists shaved off as much as 5 km/s ∆V for the mission, when compared to a simple Hohmann transfer. (Wikimedia)
I hope this clarifies the concept of a gravitational assist for you. The first two examples you gave -- Moon to escape Earth, Venus to reach Jupiter -- an accelerating assist is required and you should approach the planet from behind. For the last example -- from Venus down to Mercury -- you need to decelerate the spacecraft, and Venus should be approached head on
A couple of diagrams to show you in the clearest way possible, how gravity assists work.
The simplest form of an accelerating assist. (Courtesy Planetary Society)
Some of the possible forms of a gravity assist. (Courtesy Planetary Society)
Further Reading:
http://www2.jpl.nasa.gov/basics/grav/primer.php
http://saturn.jpl.nasa.gov/mission/missiongravityassistprimer/
http://www.planetary.org/blogs/guest-blogs/2013/20130926-gravity-assist.html
First - to answer your questions:
How should a spacecraft pass the Moon to reach escape velocity from Earth?
Well, for the most part, by the time you've reached the moon, you're likely past escape velocity from earth. The moon only moves at about 1 KM/Second, which isn't all that fast. Spacecraft usually travel in a couple to a few 10s of KMs per second. If the spacecraft is moving slower than that, it could use the moon as a gravity assist, but the moon isn't great in most cases. The 2 factors of a gravity assist are how fast the object's orbital speed is and how strong the objects gravity is. The orbital speed is what the spacecraft borrows and the planets gravity is what causes the spacecraft's angle to change. You need both for the assist to work.
How should it pass Venus for a slingshot towards Jupiter orbit?
This is a bit trickier, but basically, the spacecraft has to approach Venus from behind it's orbit and as it flies around Venus from behind (but close), it picks up speed (and Venus, correspondingly slows down, but much less cause it's far more massive - the combined orbital momentum is conserved, but like the ball bouncing off the train, the train speeds up the ball (a lot) and the ball slows down the train (a little).
Conversely, to descend Venus to Mercury?
This one, the ship flies ahead of Venus, and it slows down as it passes Venus and Venus speeds up (a little), but be careful with this one. If you slow down too much you'll fly into the sun, so this kind of slow down around Venus - as I understand it, needed to be done more gradually. That's why Messenger did 2 slow-downs around Venus and 2 around Mercury before it settled into orbit around the tiny-hot planet.
But, rather than think about it, play with the simulation.
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/Slingshot.htm
1) hit reset 2) Set the velocity and angle of approach. 3) hit start (to start Jupiter moving) 4) hit shoot - to shoot the space-ship. and don't have your speakers at full volume - in case you crash.
Watch the space ship's velocity as it goes around Jupiter. In a nutshell with this simulation, if the ship flies past Jupiter from behind it gains velocity, if it flies past Jupiter from ahead, it loses velocity.
I also like this one (which is in the link that Vedant Chandra provided), but the picture is helpful.
As far as the angle - bouncing off the train from perfect 90 degrees (from straight ahead) is the biggest gain, as the angle the ball hits the train the velocity gain decreases because that's how vector addition works.
Source (also in Vedant's link): http://www2.jpl.nasa.gov/basics/grav/primer.php
I also found this one - the website isn't in English but it might help a little, but not with angles.
and this explanation might be worth a look: http://www.schoolphysics.co.uk/age14-16/Astronomy/text/Slingshot_/index.html
and if all else fails, try this one:
Just kidding. :-)
An analogy I use to explain gravity assist to a true layman is a skateboarder and buses (the buses are old style with steel bodywork). The skateboarder has a powerful a magnet and an engine with very little fuel. The skateboarder can change direction by using her engine and modest amount of fuel, or she can use magnetic attraction on the buses going on the different routes around the area. The skateboarder can use the magnet to attract the board to the a bus that's going in a direction she wants to get pulled in. If she wants to speed up she gets behind a bus that's going the right direction and gets dragged up to the bus' speed. As she gets closer to the bus the magnetic forces get more powerful, so she steers a course that will keep her close enough to get a boost but far enough away that she doesn't get pulled into the bus. At the closest point of approach to the bus she fires her engine a little bit and propels herself away from the bus in the direction she wants to go. She's now faster than the bus.
Even with a small amount of fuel the skateboarder can get around pretty well. She can change direction or slow down as long as there are buses going the right way at the right time, that magnet will drag her towards any bus that she gets near. If there's no bus going her way though she's in for a long, slow ride though.
