# Lab 2 - Uniformly Accelerated Motion

## Introduction

All objects on the earth's surface are being accelerated toward the center of the earth at a rate of 9.81m/s2. This means that if you raise an object above the surface of the earth and then drop it, the object will start from rest and its velocity will increase by 9.81 meters per second for each second it is falling toward the earth's surface until it strikes the surface.

## Discussion of Principles

In this experiment you will measure, with the aid of computer-based instruments, the position of a falling body as a function of the time elapsed since it was released. We adopt the downward direction as positive, and we denote displacements in that direction by
y
. When air resistance is neglected, the body is said to be in free fall, and its acceleration
a
is constant.
Consider an object at position
y1
at some initial time
t1
. At a later time
t2
the object is at location
y2
. The average velocity
v12
for this object as it travels between these two points will be
( 1 )
v12 =
 (y2 − y1) (t2 − t1)

Similarly, the average velocity
v23
during the next time interval (that is, between the instants
t2
and
t3
) is
( 2 )
v23 =
 (y3 − y2) (t3 − t2)

If the acceleration is uniform or constant, the velocity at exactly the mid point of the time interval is the average velocity. This means that if the acceleration is constant, the instantaneous velocity is exactly equal to the average velocity at the midpoint of the time interval that is being examined. Even if the acceleration were not uniform, this would be a close approximation if the interval of time were short. So
v23
would occur at the midpoint of the time interval given by
( 3 )
t23 =
 (t3 + t2) 2

With these two average velocities and times we can calculate an average acceleration
a
as
( 4 )
a =
 Δv Δt
=
 (v23 − v12) (t23 − t12)

where
Δv
and
Δt
denote the change in velocity and time, respectively.
For an object moving with constant velocity, a plot of distance versus time would be a straight line with a constant slope like the graph in Fig. 1a below. Since distance is plotted on the vertical axis and time is plotted on the horizontal axis, the slope is
Δ(Distance)/Δ(Time)
or the average velocity. Here the average velocity is the same as the instantaneous velocity at any given time.
Figure 1b shows a plot of position versus time for an object moving with increasing velocity. Here we can draw a graph where we connect the points with the solid lines indicated by the red lines in Fig 1b. If we were to measure the position of the object over smaller time intervals we would see a smoother curve as indicated by the blue curve. The average velocity between two points
x1
,
t1
and
x2
,
t2
, is given by the slope of the straight line connecting these two points.
Now consider the velocity versus time version of this graph shown in Fig.2. The average velocities intersect with the instantaneous velocities at the midpoint of the two time measurements. The average of two points is the midpoint of the two points. So when we take the average of
t2
and
t3
we find the time at the half-way point. Here we refer to this time as
t23
. As shown in Fig.2, the instantaneous velocity and the calculated average velocity have the same value at this average time,
t23
. This is why we use the average time and average velocity when calculating the acceleration.
The kinematic equations are derived from the definitions of average velocity and acceleration discussed above for a uniformly accelerating object. These equations provide a useful way to evaluate the motion of an object moving with constant acceleration. For one-dimensional motion, the kinematic equations are
( 5 )
vf = vi + aΔt
( 6 )
xf = xi + viΔt +
 1 2
a(Δt)2
( 7 )
vf2 = vi2 + 2aΔx
where
vi
and
vf
are the initial and final velocities when the object is at positions,
xi
and
xf
respectively,
Δt
is the elapsed time and
a
the constant acceleration for this motion.
In summary, you can find the acceleration by considering data from two consecutive time intervals:
• 1
Calculate the average velocity
v12
for the first time interval from the distance
y2y1
traveled during the time interval
t2t1
. This is the instantaneous velocity at
t12
.
• 2
Calculate the average velocity
v23
for the second time interval from the distance
y3y2
traveled during the time interval
t3t2
. This is the instantaneous velocity at
t23
.
• 3
Calculate the acceleration
a
from the two velocities
v12
and
v23
and the elapsed time
t23t12
for these velocities.

## Objective

The objective of this experiment is to measure the position of an object in free fall as a function of time and to determine the acceleration due to gravity.

## Equipment

• Picket fence
• Photogate
• Signal Interface
• DataStudio software
• Computer
• Meter stick

## Procedure

A picket fence, which is a piece of clear acrylic (also known as Plexiglas or Lucite) covered with equally spaced black bands, will be dropped through a photogate connected to a computer. See Fig. 3. A beam of light passes from one prong of the photogate to the other. The signal interface monitors the beam and reports, to the computer, the time at which the beam was broken. From these instants of time the intervals of time between bands can be found. These time intervals, along with the distances between the bands, will allow you to calculate the average velocities of the fence during the time intervals. You can calculate the average velocity
vn,n+1
of the fence during a given time interval after measuring, the distance
yn+1yn
from the first edge of one black band to the first edge of the next black band and the time interval
tn+1tn
it took for the fence to fall that distance.

### Procedure A: Set-up and data acquisition

1
The black bands on the picket fence should be equally spaced and of equal width. Use a meter stick to measure the distance from the leading edge of the first black band to the leading edge of the second black band, as shown in Fig. 5.
2
Repeat this measurement at other locations on the picket fence and take the average value of the band width
c
, where
c = yn+1yn
, for all values of
n
. Enter this value on the worksheet.
3
Open the appropriate Pasco Capstone file for this lab. A screen similar to Fig. 6 is displayed. Note that Table 1 will be next to the Experiment Setup window.
4
You must enter the value of the picket spacing in the block labeled Flag Spacing. Enter the value in meters, and then click the Save button below.
5
Position the photogate near the edge of a table so the picket fence can fall through the photogate beam. Place a piece of clothing or similar cushioning material beneath the photogate so that the picket fence is not damaged when it hits the floor.
6
When you are ready to record data, click the Record button. See Fig. 7. Data recording will automatically begin when the photogate beam is first interrupted by the falling picket fence.
7
Position the fence just above the photogate, and release it. After the fence passes through the photogate, click the Stop button. The table, which was empty in Fig. 6, will now be filled with data that contains two columns. The first column contains the instants of time (measured in seconds) that the leading edges of the dark bands passed through the photogate and the second column gives the counts, i.e., the number of times the beam was interrupted. See Fig. 8.
Checkpoint 1:
9
After your TA has checked your work record the numbers from your Excel sheet into Data Table 1 on the worksheet.
10
Determine the average of the five acceleration values and enter it on the worksheet. See Appendix E.
11
Any object (of sufficient mass per unit volume to reduce air drag) near the earth's surface will accelerate toward the earth with the constant acceleration,
g
. The accepted value for this acceleration is 9.81 m/s2. Since the only force acting on the picket fence during its free fall was gravity, the acceleration you found should be the acceleration of gravity.
12
Compute the percent error between your average acceleration and the accepted value of the acceleration due to gravity and enter it on the worksheet. See Appendix B.

### Procedure B: Plot of velocity versus time

13
Using Excel, plot a graph of the velocity of the falling fence as a function of time. See Appendix G. Use the data from column 3 of your table for the velocities, and use column 2 for the time instants.
14
Add a linear trendline to your plot and determine the average acceleration from the slope. See Appendix H. Enter this value on the worksheet.
15
Calculate the percent error between the value of the acceleration obtained from the slope and the accepted value of the acceleration due to gravity
g
. Enter this value on the worksheet.
Checkpoint 2: