we mentioned earlier, and let’s say you wanted to determine its instantaneous velocity
at the midpoint of the window.
You could use a stopwatch to time how long it takes the egg to travel from the top to the
bottom of the window. If you then divided the height of the window by the elapsed time,
the result would be close to the instantaneous velocity. However, if you measured the
time for the egg to fall from 10 centimeters above the window’s midpoint to 10
centimeters below, and used that displacement and elapsed time, the result would be
even closer to the instantaneous velocity at the window’s midpoint. As you repeated this
process "to the limit" í measuring shorter and shorter distances and elapsed times
(perhaps using motion sensors to provide precise values) í you would get values closer
and closer to the instantaneous velocity.
To describe instantaneous velocity mathematically, we use the terminology shown in
Equation 1. The arrow and the word “lim” mean the limit as ǻt approaches zero. The
limit is the value approached by the calculation as it is performed for smaller and smaller intervals of time.
To give you a sense of velocity and how it changes, let’s again use the example of the egg. We calculate the velocity at various times using an
equation you may have not yet encountered, so we will just tell you the results. Let’s assume each floor of the building is four meters (13 ft)
high and that the egg is being dropped in a vacuum, so we do not have to worry about air resistance slowing it down.
One second after being dropped, the egg will be traveling at 9.8 meters per second; at three seconds, it will be traveling at 29 m/s; at
five seconds, 49 m/s (or 32 ft/s, 96 ft/s and 160 ft/s, respectively.)
After seven seconds, the egg has an instantaneous velocity of 0 m/s. Why? The egg hit the ground at about 5.7 seconds and therefore is not
moving. (We assume the egg does not rebound, which is a reasonable assumption with an egg.)
Physicists usually mean “instantaneous velocity” when they say “velocity” because instantaneous velocity is often more useful than average
velocity. Typically, this is expressed in statements like “the velocity when the elapsed time equals three seconds.”
v = instantaneous velocity
ǻx = displacement
ǻt = elapsed time
2.6 - Position-time graph and velocity
A graph of an object's position over time is a useful tool for analyzing motion. You see
such a position-time graph above. Values on the vertical axis represent the mouse car's
position, and time is plotted on the horizontal axis. You can see from the graph that the
mouse car started at position x = í4 m, then moved to the position x = +4 m at about
t= 4.5 s, stayed there for a couple of seconds, and then reached the position
x = í2 m again after a total of 12 seconds of motion.
Where the graph is horizontal, as at point B, it indicates the mouse’s position is not
changing, which is to say the mouse is not moving. Where the graph is steep, position
is changing rapidly with respect to time and the mouse is moving quickly.
Displacement and velocity are mathematically related, and a position-time graph can be
used to find the average or instantaneous velocity of an object. The slope of a straight
line between any two points of the graph is the object’s average velocity between them.
Why is the average velocity the same as this slope? The slope of a line is calculated by
dividing the change in the vertical direction by the change in the horizontal direction,
“the rise over the run.” In a position-time graph, the vertical values are the x positions
and the horizontal values tell the time. The slope of the line is the change in position,
which is displacement, divided by the change in time, which is the elapsed time. This is
the definition of average velocity: displacement divided by elapsed time.
You see this relationship stated and illustrated in Equation 1. Since the slope of the line shown in this illustration is positive, the average
velocity between the two points on the line is positive. Since the mouse moves to the right between these points, its displacement is positive,
which confirms that its average velocity is positive as well.
The slope of the tangent line for any point on a straight-line segment of a position-time graph is constant. When the slope is constant, the
velocity is constant. An example of constant velocity is the horizontal section of the graph that includes the point B in the illustration above.
The slope of a tangent line at different points on a curve is not constant. The slope at a single point on a curve is determined by the slope of
the tangent line to the curve at that point. You see a tangent line illustrated in Equation 2. The slope as measured by the tangent line equals
the instantaneous velocity at the point. The slope of the tangent line in Equation 2 is negative, so the velocity there is negative. At that point,
Position-time graph
Shows position of object over time
Steeper graph = greater speed