The meaning of velocity

Intro to the mathematics of motion

Today, we are going to talk about how we think about speed and the rate of change in speed (usually called acceleration).  It is a bit math-y, but don't panic - we'll summarize things nicely in a couple of simple-to-use equations.

First, let's look at some definitions.

Average (or constant) velocity, v

v = d / t

That is, distance divided by time.  The SI units are meters per second (m/s).

* Strictly speaking, we are talking about speed, unless the distance is a straight-line and the direction is also specified (in which case "velocity" is the appropriate word).  However, we'll often use the words speed and velocity interchangeably if the motion is all in one direction (1D).

Some velocities to ponder....

Approximately....

Keep in mind that 1 m/s is approximately 2 miles/hour.

Your walking speed to class - 1-2 m/s
Running speed - 5-7 m/s
Car speed (highway) - 30 m/s
Professional baseball throwing speed - 45 m/s
Terminal velocity of skydiver - 55 m/s
Speed skiing - 60 m/s
Speed of sound (in air) - 340 m/s
Bullet speed (typical) - 900 m/s
Satellite speed (in orbit) - 6200 m/s
Escape velocity of Earth - 11,200 m/s
(That's around 7 miles per second, or 11.2 km/s)


What about.....
The Speed of light

Speed of light (in a vacuum) -

c = 299,792,458 m/s

This number is a physical constant, believed to be true everywhere in the universe. The letter c is used to represent the value being of constant celerity (speed).

By the way, it's hard to remember this exact number, and I wouldn't expect you to.  However, here are some approximations that may make it easier to keep it in mind.  The speed of light is approximately:

- 300,000,000 meters/sec

- 186,000 miles/sec

- 7 times around the Earth's equator in 1 second

- Out to the Moon in around 1 second (1.3 seconds is closer) - so, the Moon is approximately 1.3 "light seconds" away (on average)

- To the Sun in about 8 minutes - so, the Sun is approximately 8 "light-minutes" away (on average)

- To Mars in about 13 minutes, though this varies depending on the relative locations of Earth and Mars in their respective orbits.  (Consider that this is, under the best circumstances, about a 9 month one-way trip at current spaceship speeds.)

- To the nearest (non-Sun) star, Proxima Centauri B (actually part of a 3-star system, and home to a recently-discovered Earth-like planet!), in 4.24 years.  Yes, YEARS.  So, that 3-star system is around 4.3 "light years" away from us.  And that's our closest neighbors!!  See why we don't get too far in space travel?

Instantaneous Velocity

Average velocity should be distinguished from instantaneous velocity (what you get, more or less, from a speedometer):

v(inst) = d / t, where t is a very, very, very tiny time interval. There's more to be said about this sort of thing, and that's where calculus begins.

Now the idea of velocity is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip. However, if you care about the details of velocity, if and when it changes, then we need to introduce a new concept: acceleration.