Orbits of satellites and moons
This sequence of ideas can help students understand orbital motion.
Throw a ball out horizontally. It falls to the ground some distance away. A rifle bullet, fired faster but also horizontally, reaches the ground after a kilometre or so.
A thought experiment: fire a bullet so fast that it covers an appreciable part of the Earth’s circumference before it reaches the ground. What effect does the Earth’s curvature have on the bullet’s fate? Fire it even faster and it ‘falls over the edge of the Earth’. The Earth falls away from the bullet’s original direction at exactly the same rate as the bullet falls. To the bullet, all parts of the Earth are the same. It soon forgets where it started from. With just the right speed, it will always be falling over the edge and so it will go on round the Earth- keeping just above the ground - until it arrives back at the starting point and hits us from behind.
The bullet started off as a projectile with a constant horizontal velocity and a vertical acceleration due to gravity. Therefore at every point in its orbit the Earth satellite must have a constant acceleration towards the centre of the Earth.
Newton imagined that if he threw the stone fast enough it would orbit the Earth because it is always falling towards the Earth at the same rate as the Earth "falls away" from it. The projectile becomes a satellite.
This thought experiment is sometimes known as Newton’s cannon and is available as a computer simulation or video.
In practice air resistance absorbs energy and down comes the bullet, so you must start outside the atmosphere. When a rocket is used to launch a satellite, the motions of rocket and satellite can be analyzed like this:
• The rocket starts off nearly vertically. The exhaust gases exert an upward push greater than the rocket’s weight, so that the rocket accelerates upwards.
• Fuel exhausted; motor cuts out; first stage jettisoned.
• Parabolic (free-fall trajectory until the path is horizontal at maximum altitude.
Final stage ignites and accelerates its relatively small mass to high velocity. Satellite unlatched and left in orbit. Final stage is also in orbit but at a slower speed so it gets left behind.