Physics presentation: speed and velocity

Planes, planes, and more planes. Oh wait, that's just the same plane, isn't it? (Movie link: "Paths of Flight.")

Let's introduce the different types of (one-dimensional) velocities used to describe motion along a line.

We'll first need to define two different types of "paths," as somehow measured from an initial starting point to a final ending point.

Distance traveled is conventionally how far one would travel, regardless of direction, even going back-and-forth. This is always a positive quantity, as even moving backwards is counted as distance traveled.

Displacement is the straight-line distance from the initial starting point, to the final ending point--"as the crow flies" (but not necessarily how the pigeon walks). This can be positive or negative, depending on the direction from the initial to the final location, for horizontal displacements we will use positive values to indicate displacements that point to the right, and negative values to indicate displacements that point to the left. This is always a straight line, regardless of the back-and-forth distance traveled while en route from the initial starting point to the final ending point.

In general, distance traveled will be __________ the magnitude of displacement.
(A) less than.
(B) equal to.
(C) greater.
(D) (More than one of the above choices.)
(E) (None of the above choices.)
(F) (Unsure/guessing/lost/help!)

With these two types of paths defined, now let's consider speed, and as it turns out, two different types of velocity.

Consider the contrail left by this airplane--definitely not straight-line travel. Estimating the distance along this curving path, and dividing that distance traveled by the time interval spent traveling along this path...

...would result in the average speed of the aircraft, as it traveled along this path. Note that this is always a positive quantity.

This is an actual view from Google Maps (courtesy of Google Sightseeing). The same plane is shown in several different locations, as seen from a satellite overhead, as it was captured on consecutive vertical passes of the imaging camera. Knowing the displacement--the straight-line distance from the initial starting point to the final ending point--and dividing that by the time interval spent traveling along this displacement...

...would result in the average velocity of the aircraft, as it traveled along this displacement. Note that this can be either a positive or a negative quantity, depending on the direction of the displacement. (Some textbooks draw a horizontal "bar" over the "v_x" to denote that it is an average quantity, which could be confused with a vector arrow--here we denote an average quantity by labeling it in the subscript, in order to avoid this confusion.)

In general, average speed will be __________ the magnitude of average velocity.
(A) less than.
(B) equal to.
(C) greater than.
(D) (More than one of the above choices.)
(E) (None of the above choices.)
(F) (Unsure/guessing/lost/help!)

Another actual view from Google Maps (again, courtesy of Google Sightseeing). Note the detailed monochrome image of these two planes, and the matching corresponding blurry color image for each plane. This is due to the satellite camera making two separate monochrome and color images on the same pass, but separated by a very brief time interval. In this case, taking the straight-line displacement between the initial starting point to the final ending point is correspondingly small--and dividing that by the brief time interval spent traveling along this displacement...

...would result in what would approach the instantaneous velocity of the aircraft, as it traveled along this small displacement during a brief time interval. (Note again that this can be either a positive or a negative quantity, depending on the direction of the displacement.) To properly make this an instantaneous velocity, we would do some calculus mumbo-jumbo where the limit of a vanishingly small time interval is taken in dividing into a correspondingly infinitesimal displacement.

So we have average speed, average velocity, and instantaneous velocity (although we often leave out "instantaneous" when referring to velocity, as that is implied, but strive to always explicitly leave in the "average" terminology).

In general, which of the following quantities could be negative?
(A) Average velocity.
(B) Average speed.
(C) (Instantaneous) velocity.
(D) (Instantaneous) speed.
(E) (More than one of the above choices.)
(F) (All of the above choices.)
(G) (None of the above choices.)
(H) (Unsure/guessing/lost/help!)

What about (instantaneous) speed? How is it related to (instantaneous) velocity?

In general, (instantaneous) speed will be __________ the magnitude of (instantaneous) velocity.
(A) less than.
(B) equal to.
(C) greater than.
(D) (More than one of the above choices.)
(E) (None of the above choices.)
(F) (Unsure/guessing/lost/help!

What about your car's odometer--what quantity does that measure?

An odometer measures an object's:
(A) displacement.
(B) distance traveled.
(C) (instantaneous) velocity.
(D) (instantaneous) speed.
(E) (Unsure/guessing/lost/help!)

And your car's speedometer--what quantity does that measure?

A speedometer measures an object's:
(A) displacement.
(B) distance traveled.
(C) (instantaneous) velocity.
(D) (instantaneous) speed.
(E) (Unsure/guessing/lost/help!)