Thinking about falling
Lines vs Circles
Remember Aristotle's distinction between natural motion (falling down towards the earth) and celestial motion (circular motion) of the stars in the night sky?
Apples vs the Moon
His campus closed because of disease, Isaac Newton returns to his family's farm. One day he sees an apple falling while the moon was in the sky and wonders,
Is the moon "falling" too?
You must generally aim just above your target, to allow for gravity acting on your water stream / bullet / whatever you throw.
The precise mathematical form of this trajectory near the surface of the earth is a parabola, a particularly simple mathematical equation has this shape: $y= -a x^2$.
The monkey drop
Special case: when your target drops at the same moment that you shoot. Both target and, er, um, tranquilizer dart will drop at the same rate $\Rightarrow$ aim straight at target.
The moon "falling" around the earth.
Earth's surface falling away
As you throw faster, the ball lands farther away, and you start to have to worry about the roundness of Earth.
Ask about the forces (acceleration) on the ball at the pictured points for the slower and faster throw.
At 29,000 km/hr ~ 8 km / s the earth is "falling away" just as fast as your ball is "falling down", and we say an object is "in orbit".
You can think of the moon's motion in this way too: It is always falling towards Earth, but it has a lot of motion to the side too, so it misses again and again.
The International Space Station orbits about 300 km above the surface of Earth. Is there no gravity there?
In a bit we'll see how to calculate the force of gravity. It turns out that at this height gravity still works, and the force of gravity is about...90% of what it is on Earth's surface.
So are these guys just deluded?
These guys experience somewhat the same thing. Or the famous "Vomit Comet" is an aircraft that follows a flight path designed to "fall" in such a way as to simulate the environment in a space station.
The force of gravity is weaker as you go further away, It also takes longer to go around the earth as you go further out.
At a height of ~36,000 km about the surface of the earth (radius ~6,400 km) the time it takes a satellite to orbit the earth is precisely one day.
If the satellite is positioned above the earth's equator, it would appear to hover always at the same place in the sky.
These orbits are also used for telecomm satellites. You can point at one, and it's always in the same place.
This is too far for spy satellites.
Conceptual Exercises in Chapter 4: 21, 23, 24.