Velocity

Note that, as with the position, the velocity is a vector: it
has a magnitude and a
direction associated with it. For example, if someone travels from Toronto to
Vancouver by airplane in three hours, their average velocity for that
trip is 3000 km divided by 3 hours, or 1000 km/hr, due east. Sometimes
only the magnitude of the velocity is of interest. It is called the
**speed**. The average speed for the above trip from Toronto to
Vancouver is 1000 km/hr. No direction need be specified.
Note that objects moving with a large speed
change their position quickly over time, whereas those moving with a
small speed take longer to cover the same distance.

The average velocity only depends on the initial and final points of the
travel - it knows nothing of the details of the motion between these
points. For this reason one introduces the concept of
**instantaneous velocity**, which applies to a single point, or
time of the
motion. The instantaneous velocity is defined as follows. Suppose one wants to define the
instantaneous velocity at a certain point, say **A** along the journey.
One first calculates the
average velocity between this point and an arbitrary nearby point, say
**B**_{1}. The arrow in the Figure below pointing from **A** to
**B**_{1} indicates the direction of the average velocity in going
between those two points. One then
does the same, but for a point *B*_{2} which is nearer to the initial point
**A** of interest. Then again for *B*_{3} which is closer
still, and so on. The instantaneous velocity is obtained by repeating
this process until you get to a point **B** that is arbitrarily
close to **A**.

Thus the definition of instantaneous velocity is:

As the Figure indicates, as the **B** gets closer and closer to **A**
the direction of the velocity (indicated by the arrows) gets closer
and closer to pointing along a line that is **tangent** to the trajectory
at the point **A** (i.e. points along the trajectory).
The **instantaneous speed** is correspondingly defined as the
magnitude of the instantaneous velocity.
An example with which we are all familiar involves travel by car.
Suppose we are driving between Winnipeg and Kenora. This road is more or
less completely straight, so we don't have to worry about directions. After two hours we stop for a one hour lunch break, and then continue
to drive for another hour. At this stage, the odometer shows that we have driven 300 km, so that
our average speed for that portion of the trip would be 300/4 = 75 km/hr. The magnitude of our instantaneous speed, on the other
hand is given by the speedometer reading, and can be very different
from the average speed. For example we might be going at 100 km/hr
or if we happen to be stopped at a traffic light, our instantaneous speed might be zero.
In effect, the odometer counts the total number of revolutions made
by the tires (and multiplies by the circumference of the tire) in
order to get the total distance traveled. The speedometer instead
counts the number of revolutions in a very short time interval, say
half a second, in order to tell us how fast the car is going at that
instant.

1999-09-29