12.8 - Interactive problem: Newton’s cannon
Imagine that you are Isaac Newton. It is the year 1680, and you are staring up at the
heavens. You see the Moon passing overhead.
You think: Perhaps the motion of the Moon is related to the motion of Earth-bound
objects, such as projectiles. Suppose you threw a stone very, very fast. Is there a
speed at which the stone, instead of falling back to the Earth, would instead circle
the planet, passing around it in orbit like the Moon?
You devise an experiment in your head, a type of experiment called a thought
experiment. A thought experiment is a way physicists can test or explain valuable
concepts even though they cannot actually perform the experiment. You ask: “What
if I had an extremely powerful cannon mounted atop a mountain. Could I fire a stone
so fast it would never hit the ground?”
Try Newton’s cannon in the simulation to the right. You control the initial speed of
the cannonball by clicking the up and down buttons in the control panel. The cannon
fires horizontally, tangent to the surface of the Earth. See if you can put the stone
into orbit around the Earth. You can create a nearly perfect circular orbit, as well as
orbits that are elliptical.
(In this simulation we ignore the rotation of the Earth, as Newton did in his thought experiment. When an actual satellite is launched, it is fired in
the same direction as the Earth’s rotation to take advantage of the tangential velocity provided by the spinning Earth.)
12.9 - Circular orbits
The Moon orbits the Earth, the Earth orbits the Sun, and today artificial satellites are
propelled into space and orbit above the Earth’s surface. (We will use the term satellite
for any body that orbits another body.)
These satellites move at great speeds. The Earth’s orbital speed around the Sun
averages about 30,000 m/s (that is about 67,000 miles per hour!) A communications
satellite in circular orbit 250 km above the surface of the Earth moves at 7800 m/s.
In this section, we focus on circular orbits. Most planets orbit the Sun in roughly circular
paths, and artificial satellites typically travel in circular orbits around the Earth as well.
The force of gravity is the centripetal force that along with a tangential velocity keeps
the body moving in a circle. We use an equation for centripetal force on the right to
derive the relationship between the mass of the body being orbited, orbital radius, and
satellite speed. As shown in Equation 1, we first set the centripetal force equal to the
gravitational force and then we solve for speed.
This equation has an interesting implication: The mass of the satellite has no effect on
its orbital speed. The speed of an object in a circular orbit around a body with mass M
is determined solely by the orbital radius, since M and G are constant. Satellite speed
and radius are linked in circular orbits. A satellite cannot increase or decrease its speed
and stay in the same circular orbit. A change in speed must result in a change in orbital
radius, and vice versa.
At the same orbital radius, the speed of a satellite increases with the square root of the
mass of the body being orbited. A satellite in a circular orbit around Jupiter would have
to move much faster than it would if it were in an orbit of the same radius around the
Earth.
Circular orbits
Satellites in circular orbit have constant
speedOrbital speed and radius
Satellite speed and radius are linked
·The smaller the orbit, the greater the
speed