Monday night I got to see the lovely sight of Venus and the thin crescent Moon shining close to each other in the dusk after sunset. I was somewhat surprised when I saw them again half an hour later, only a little closer to the western horizon. And then yet again, some fifteen minutes after that. Living in the tropics as I do, I'm used to things setting (and rising) very quickly. If you see something close to the horizon, you'd better look quick, because it'll be gone before too much longer. Here in California, further from the equator, things hang around longer before dropping belong the horizon.
The reason has to do with the fact that the Earth is a sphere (to a good first approximation). If you're in the tropics (between \(\pm\)23.5\(^\circ\)), the Sun appears to set very nearly perpendicularly to the horizon. Not only that, but your tangential speed \(-\) the speed you travel around the circle you proscribe on the surface of the Earth each day \(-\) is higher than it is further north or south, since you're further from the rotational axis of the Earth. As you travel away from the equator, celestial objects appear to set at more of an angle, and your tangential speed is lower as well, leading to objects taking longer to set (or rise).
Some simple mathematical calculations show that the Sun, about half a degree on the sky, would take about two minutes to set if you were located on the equator and it was either the spring or fall equinox. As you travel further from the equator that time increases until you reach the Arctic or Antarctic circles, where upon the time to set ends up being longer than 24 hours, and the Sun simply travels around the sky. Above the Arctic circle (or below the Antarctic one) you are, in theory, guaranteed at least one, 24-hour period in which the Sun is above the horizon the whole time, and one 24-period in which the Sun is below the horizon the entire time. In theory, these would occur on the summer and winter solstices for the norther hemisphere, vice-versa in the southern. (In practice this depends on other factors such as clarity of the atmosphere, height of the observer above sea level, and the fact that the Sun is a disk on the sky and not a point.)
If you're wondering about similar considerations for the Moon, the picture is fairly similar, albeit modified by two important considerations: one, the Moon's orbit is inclined from the plane of the Earth's equator; and two, the Moon orbits the Earth, and so has a motion of its own that slows down its rising and setting (since it appears to move eastward across the sky, opposite the apparent direction given it by the rotation of the Earth). The Moon is nearly the same size as the Sun on the sky so a naïve calculation would give it about two minutes to set as a minimum time, but in practice the combination of all the factors mentioned above means it will always take longer than that. (Of course, since the full Moon sets at dawn, I doubt too many people are interested in how long it takes to set.)
A hui hou kākou!