Theory of gravity and theory of motion (a.k.a. "Newton's laws") developed simultaneously, drawing on astronomical observations and terrestrial experiments.
For simplicity of explanation, we will treat motion first, then
turn to gravity.
Velocity is the rate of change of position. For an object moving in a straight line at fixed speed:
But velocity has a magnitude and a direction.
In mathematical terms, velocity is represented by a vector (roughly speaking, an arrow).
The magnitude of the velocity (length of the vector) is called the speed.
Objects moving with the same speed in different directions have different velocities.
For motion in one direction, we can specify forward (positive velocity)
and backward (negative velocity).
For three dimensions, we need to specify the x-velocity ("east-west"), the y-velocity ("north-south"), and the z-velocity (up-down).
Acceleration is the rate of change of velocity.
It is also a vector, with a magnitude and a direction.
If you start from rest, the direction of the acceleration tells which direction you will go, and the magnitude tells how rapidly you will reach a certain speed.
For motion in a straight line, starting from rest, with constant
acceleration for time t,
velocity at beginning = 0
velocity at end = a x t.
average velocity = 1/2 (a x t).
distance traveled = (avg. velocity) x t = 1/2 (a x t) x t = 1/2 at2.
Note that in this equation, a represents acceleration, not
the semi-major axis of a planet's orbit.
In physics, we often quote distances in meters (m) and time in
seconds (sec, or sometimes just s).
Since velocity = distance/time, the corresponding units of velocity are meters-per-second (m/sec).
Since acceleration = (change of velocity)/time, the corresponding units of acceleration are meters-per-second-per-second, usually written (m/sec2).
For example, the acceleration of falling objects near the surface of
the Earth is 9.8 m/sec2.
If an object starts at rest and falls for one second, its velocity is
In astronomy, it is sometimes convenient to use other units for
distances, velocities, and accelerations.
For example, we may describe the velocities of planets in km/sec (1 km/sec = 1000 m/sec) or in AU/year.
The orbital velocity of the Earth is 2\pi AU/year = 30 km/s.
(Since the Earth's orbit is not perfectly circular, this is really an average velocity, with variations of a few percent.)
Going in a circle at constant speed requires acceleration.
A: Because the direction of the velocity is changing, even though the magnitude (speed) is not.
Consider an object moving at speed v in a circle of radius r.
Q: As it goes halfway around the circle, how much does its velocity change?
A: From +v (forward) to -v (backward): change is 2v.
Q: How long does it take to go halfway around?
A: distance = \pi x r. Therefore, time = distance/velocity = \pi x r / v.
The average acceleration over this time is
One can show that the instantaneous acceleration in circular motion is
Galileo: Force causes acceleration.
Q: In what direction do you need to apply force to keep an object moving in a circle at constant speed?
A: Toward the center of the circle.
Analogy: whirling an object on a string.
Q: What happens if there is no force?
A: The object continues on in a straight line.
Analogy: let go of the string!
Conclusion: Planets can orbit in circles around the Sun if the Sun
exerts a force that pulls them toward it.
Planets are always falling toward the Sun, but because they are moving, they "miss" and continue in a circle.
Explaining the motions of planets requires a force that attracts them to the Sun.
Similarly, the Moon is always falling towards the Earth.
More precisely, a planet moves in an ellipse, and its speed increases
as it gets closer to the Sun.
The principles are the same as for circles, but with an ellipse, the attractive force can change the speed of the planet along its path as well as changing the direction of the path.
The Moon also moves in an ellipse around the Earth, and it also obeys Kepler's equal area rule.
In Isaac Newton's Principia (1687), he summarizes his theory of mechanics with three "laws":
First Law: A body remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force.
Second Law: Force on a body causes it to accelerate in the direction of the force. The acceleration is proportional to the strength of the force and inversely proportional to the mass of the body.
Third Law: Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.
The first law is the principle of inertia, a (much) more clearly stated version of the discovery of Galileo and his predecessors.
The third law is sometimes phrased: "For every action, there is an equal and opposite reaction."
Note that Newton's "laws" really constitute a theory of
motion and forces.
This theory is highly successful in explaining a wide range of phenomena.
Newton's second law is best summarized by a fundamental equation:
m = mass
a = acceleration
Note that this is equivalent to
F and a are really vectors, with
a magnitude and a direction.
m is just a number, with no direction.
The mass of a body is, effectively, its "amount of stuff."
It can be measured, from Newton's second law, by measuring the body's resistance to acceleration.
The standard, metric system unit of mass is the kilogram (kg).
The weight of a body is the force that gravity exerts on it.
For objects on the Earth, the weight is proportional to the mass.
The same object on, say, the Moon, would have the same mass, but it would weigh less (1/6 as much).
The standard, metric system unit of force is the newton.
Recall that, on Earth, acceleration under gravity is 9.8 m/sec2.
The weight of a 1 kg object is
In English units, a 1 kg object weighs about 2.2 pounds.
Thus, 1 newton = 2.2/9.8 pounds = 0.225 pounds.
First law, inertia:
A bowling ball is rolling towards me. Why do I have to push to stop it?
A tricky question: Why does the ball eventually stop even if I don't push it?
Second law, F=ma:
Which is harder to get moving, a billiard ball or a bowling ball?
Do I push harder to get something moving slowly, or to get it moving fast?
Third law, action and reaction:
I step off of a rowboat onto the shore. What happens to the boat?
Standing on a skateboard, I fire a bullet from a high powered rifle. What happens to me?
Second and third law combined:
A 500-pound man and I collide on our skateboards. Who bounces back further?
Back to the gun: Why don't I fly backwards as fast as the bullet flies forward?