In our discussion so far we have made a distinction between
velocity and speed, where speed is the absolute value of the rate
of change of displacement with respect to time and velocity
includes both a magnitude
and direction. In the 1 dimensional case we have here, the only
direction information required is whether the motion is towards
increasing position numbers (positive), or towards decreasing
position numbers (negative). Still, some direction is given by
the algebraic sign so we call the rate of change of displacement
with respect to time, velocity.
Remember from our
rate of change
discussion that the rate of change of one variable with respect
to another is the ratio of the change in the dependent variable
to the corresponding change in the independent variable. What
that means in our case is that the rate of change of position
with respect to time between P_{1} and P_{2} is
(x_{2}  x_{1}) / (t_{2}  t_{1}).
So
Dx = x_{2}  x_{1}
and
D t = t_{2}  t_{1}
In this special case where x represents position, Dx is the displacement.
Because we knew the position and time at two instances, we were
able to calculate a velocity
v = Dx / Dt .
To indicate that this velocity is only the average velocity
between position x_{1} and x_{2}, we symbolize it
with a bar over the v, like this , so
= Dx / Dt .
In general if we use the subscript _{i} to denote initial
conditions and the subscript _{f} to denote final
conditions, then
Dx = x_{f}
 x_{i}
and
Dt = t_{f}
 t_{i} ,
so
= Dx / Dt = (x_{f}  x_{i}) /
(t_{f}  t_{i}) .
Be careful in the order in which you
subtract one value from another to get the
D of a variable. If you are inconsistent, the sign of the
velocity will come out wrong. Most folks subtract the initial
value from the final value. This is a convention that helps us
keep our signs straight. It is a good practice to use, even if
other schemes might work.
Let's calculate the average velocity between t_{1}
and t_{2} from the data we were given. That gives us
= (133) / (161) = 10/15 = .6667 m/s
(meters per second) ,
to 4
significant figures.
You may have become accustomed to answers that come
out even. In physics, whatever happens... happens.
Now we know from our work with rates of
change that the instantaneous rate of change of position with
respect to time is the slope of the position vs. time plot, so
instantaneous velocity is the slope of the position vs. time plot
at the instant in time we are considering. In the
Position & Velocity vs. Time
display we
have measured the slope of the position vs. time plot at many
points along the time axis and plotted the results.
