12.7 - Newton’s cannon
In addition to noting that the Earth exerts a force on an apple, Newton also pondered
why the Moon circles the Earth. He posed a fundamental question: Is the force the
Earth exerts on the Moon the same type of force that it exerts on an apple? He
answered yes, and his correct answer would forever change humanity’s understanding
of the universe.
Comparing the orbit of the distant Moon to the fall of a nearby apple required great
intellectual courage. Although the motion of the Moon overhead and the fall of the apple
may not seem to resemble one another, Newton concluded that the same force dictates
the motion of both, leading him to propose new ways to think about the Moon’s orbit.
To explain orbital motion, Newton conducted a thought experiment: What would happen
if you used a very powerful cannon to fire a stone from the top of a very tall mountain?
He knew the stone would obey the basic precepts of projectile motion, as shown in the
diagrams to the right.
But, if the stone were fired fast enough, could it just keep going, never touching the
ground? (Factors such as air resistance, the Earth’s rotation, and other mountains that
might block the stone are ignored in Newton’s thought experiment.)
Newton concluded that the stone would not return to the Earth if fired fast enough. As
he wrote in his work, Principia, published in 1686:“ ... the greater the velocity with which [a stone] is projected, the farther it goes
before it falls to the earth. We may therefore suppose the velocity to be so
increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it
arrived at earth, till at last, exceeding the limits of the earth, it should pass into
space without touching.”Newton correctly theorized that objects in orbit í moons, planets and, today, artificial
satellites í are in effect projectiles that are falling around a central body but moving fast
enough that they never strike the ground. He could use his theory of gravity and his
knowledge of circular motion to explain orbits. (In this section, we focus exclusively on
circular orbits, although orbits can be elliptical, as well.)
Why is it that the stone does not return to the Earth when it is fired fast enough? Why
can it remain in orbit, forever circling the Earth, as shown to the right?
First, consider what happens when a cannon fires a cannonball horizontally from a
mountain at a relatively slow speed, say 100 m/s. In the vertical direction, the
cannonball accelerates at g toward the ground. In the horizontal direction, the ball
continues to move at 100 m/s until it hits the ground. The force of gravity pulls the ball
down, but there is neither a force nor a change in speed in the horizontal direction
(assuming no air resistance).
Now imagine that the cannonball is fired much faster. If the Earth were flat, at some
point the ball would collide with the ground. But the Earth is a sphere. Its approximate
curvature is such that it loses five meters for every 8000 horizontal meters, as shown in
Concept 2. At the proper horizontal (or more properly, tangential) velocity, the
cannonball moves in an endless circle around the planet. For every 8000 meters it
moves forward, it falls 5 meters due to gravity, resulting in a circle that wraps around the
globe.
In this way, satellites in orbit actually are falling around the Earth. The reason
astronauts in a space shuttle orbiting close to the Earth can float about the cabin is not
because gravity is no longer acting on them (the Earth exerts a force of gravity on
them), but rather because they are projectiles in freefall.Newton’s cannon
Newton imagined a powerful cannon
The faster the projectile, the farther it
travels
At ~8,000 m/s, projectile never touches
groundClose-up of Newton’s cannon
At high speeds, Earth’s curvature affects
whether projectile lands
At ~8,000 m/s, ground curves away at
same rate that object fallsObjects in orbit
Move fast enough to never hit the
ground
Continually fall toward the ground,
pulled by force of gravity(^228) Copyright 2000-2007 Kinetic Books Co. Chapter 12