**PRELAB**

**PURPOSE**

You will study how the average velocity approaches a fixed value as the time interval over which it is measured approaches zero.

**EQUIPMENT**
air track, glider, two photogates, shims

**RELEVANT EQUATIONS**

Average Velocity | ||

Instantaneous Velocity |

**DISCUSSION**

The concept of the instantaneous
velocity as the limiting value of the average velocity taken over smaller
and smaller time intervals is important in the study of motion. When the
velocity is constant at all times, then the average value of the velocity,
as defined in the first equation above will give the same value no matter
how large or small the time interval is. However, if the velocity is changing
from instant to instant, and we examine the average value centered about
a specific time **t**, then the ratio of the distance covered
**Δx**, divided by the time interval **Δt**,
depends on the width of the time interval **Δt** over which the
average is taken. It is an amazing consequence of the limiting
process that if the time interval around time **t** is successively
reduced, then the average velocity approaches a fixed limiting value. We
can define this limiting value to be the __instantaneous velocity__
at time **t**. The limiting process is illustrated in **Figure 6-1**,
where the average velocity is seen to be the slope of the straight line
connecting points **A** and **B**. As we squeeze down on the interval
about time **t**, this slope, and thus the average velocity,
approaches a fixed value.

An air track provides a nearly frictionless surface for one-dimensional motion of a glider. If the track is level and the glider is given a slight nudge, it will move with constant velocity along the length of the track. Since the velocity is constant, the average velocity measured between any two points on the track is equal to the instantaneous velocity of the glider.

**Figure 6-1: Air Track Set-up
for Measurement of Instantaneous Velocity**

Referring to **Figure 6-1**, if
points **A** and **B** are selected, the average velocity is:

where: **Δx**
is the distance from **A **to **B** and **Δt**
is the time for the glider to move from **A** to **B**.

If the track is tilted slightly
so that the glider accelerates from **A** to **B**, the velocity
will be different at every point between **A** and **B**. The velocity
increases at a constant rate due to gravity as the glider moves toward
**B**.
The average velocity measured between **A** and **B** ()
will be less than the instantaneous velocity at point **B**,
v_{B}(t),
because the velocity is increasing as it moves towards point **B**.

To experimentally estimate
v_{B}(t), you can use the definition of instantaneous velocity:

As you move **A** closer to **B**
many times, each new value of
approximates v_{B}(t), better, but it is still
less than v_{B}(t). Since it may be impractical
to make **Δx**, and hence **Δt**, small enough for
to equal v_{B}(t), you can extrapolate to
**Δt** = 0 on a graph of
versus **Δt**

It may have already occurred to
you that a better estimate of v_{B}(t) could
be made using points on each side of **B** (e.g. **A** and **C**
of** Figure 6-1**). Since the velocity is less than v_{B}(t)
on the "**A** side" of **B** and greater than v_{B}(t)
on the "**C** side" of **B**,
should produce a better approximation to v_{B}(t)
than the one sided procedure. You can move both **C** and **A** closer
to **B** and calculate new values of
for successively smaller values of **Δx** and **Δt**.
These
values usually converge to v_{B}(t)
faster than the values. A graph of
versus **Δ** can still help you extrapolate to **Δt** =
0. You will use this three-point procedure to estimate an air track glider's
instantaneous velocity at a point **B** midway between measuring points
**A** and **C**.

Print out and complete the
**Prelab questions**.