The Kinematics of Free Fall
1

1
1
Kinematics of Free Fall
1.1
Introduction
Kinematics is the
description
of the motion of material objects. The description of a
moving
object is a record of the
position
of the object as a function of time. Once this
record of position versus time is obtained, it can be used to obtain the
velocity
of the
object (the rate of change of position) and the
acceleration
of the object (the rate
of
change of velocity). In this lab we study the kinematics of an object that is undergoing
two particular kinds of motion, constant velocity and “free fall”. Constant velocity
happens when there are no outside forces present on the object being observed.
Free fall
motion happens when the only force acting on the object is the gravitational attraction of
the Earth. However, neither of these two happens on Earth because of air resistance.
When an object moves through the air it experiences a “drag” force
which opposes its
motion. This drag force is proportional to the squared speed of the object and its cross
sectional area perpendicular to the motion. The purpose of this experiment is to show that
in the horizontal direction there is not significant accel
eration and to measure the
acceleration due to gravity during free fall in the vertical direction.
The object we will use is a small steel sphere (its size will help reduce air resistance).
A “webcam” will also be used to take pictures of the steel ball
as it moves through the
air. A grid placed behind the steel ball will enable us to see the position of the steel ball
in each frame and the repetition rate of the webcam will enable us to get the time that
each frame happened at.
The major errors in this
lab will be due to the resolution of the camera and thus in the
ability to accurately find the position of the steel ball as a function of time.
1.2
Background Discussion
Most likely in your education you learned the
simple equation:
time
velocity
D
(1.0)
This equation is a very useful equation but it
only works under limited conditions; namely,
constant velocity and a starting position set at a
distance of zero.
Let us now consider a slightly more
complicated set of conditions. Assume
a small mass
moves freely under the influence of a constant force
(constant acceleration) (see Fig. 1

2). This will cause the velocity to not remain constant,
however, let us assume that we know the initial conditions (initial velocity,
v
i
and initial
posi
tion,
x
i
) If we are able to record the final position,
x
f
, and the time,
f i
t t t
, it took
The Kinematics of Free Fall
1

2
to travel between the starting and ending positions we can related the initial conditions to
the acceleration as followed:
2
)
(
2
1
t
a
t
v
x
x
i
i
f
(1.2)
and
t
a
v
v
i
f
(1.3)
where
a
is the acceleration and
v
f
is the velocity as it passes the final position. It should
be noted that equations (1.2) and (1.3) are only valid if the acceleration is constant
durin
g the entire time interval. We can see that eqations (1.2) and (1.3) reduce to
equation (1.1) if the acceleration is zero. Finally, if the only force is due to gravity then
we normally use the variable “
g
” to represent the acceleration due to gravity (in
place of
“
a
”
and again, only if the force of gravity is constant
).
It is useful to plot the motion of the mass in a
velocity versus time plane. This is done in Fig. (1

3) which shows the velocity,
v
i
at time
t
i
and the
velocity
v
f
at time
t
f
. The avera
ge acceleration of the
particle is
,
f i
f i
v v
v
a
t t t
(1.4)
where
.
f i
v v v
. This is just the slope of the
line between points, (
v
i
, t
i
) and (
v
f
, t
f
). If we
measure
v
in m/s and
t
in seconds, the
average acceleration will be given in m/s
2
. Thus,
if the instantaneous acceleration is the same at
every point throughout the motion (acceleration is
constant), it is also equal to average accelera
tion
of the particle.
The average velocity,
v
, during the time interval, t
i
to t
f
, is equal to the
instantaneous velocity, v
m
, at the midpoint time of the interval,
f i
t t t
,
if the
acceleration is constant. T
hese expressions will be very useful in analyzing our
experiment and enables us to find our midpoint velocities by finding the average
velocities over the time interval. In other words
m
v v
(1.5)
The average velocity is found using
a standard average formula and combining with
equation (1.4) we see that
m
f
i
v
v
v
v
2
(1.6)
The Kinematics of Free Fall
1

3
The midpoint time can be figured out using the same standard averaging formula
2
f
i
m
t
t
t
(1.7)
We also know tha
t the velocity of an object is just the change of distance divided by
the change of time.
i
f
i
f
t
t
x
x
t
x
v
(1.8)
Combining all of these formulas enables us to know a set velocities and
corresponding times for an object if we know a set of positions
and corresponding times
and that is undergoes constant acceleration.
There are two more things to say on
acceleration, velocity, and displacement
. The
first is the “x” variable is normally used for horizontal displacement (position) but the
same equations
work for a vertical displacement (designated normally by “y”).Second,
acceleration, velocity and displacement
are technically vector quantities (we will learn
more about this in lab 2) so what happens in the horizontal direction is independent of
what hap
pens in the vertical direction. The second experiment we do will help to show
this by showing that there is
acceleration
in one direction and no acceleration in the other
direction.
1.3
Description of Experiment
The setup for this lab is a very standard way o
f determining velocities using a
camera with a knowing frame rate and a background that has known distances.
The camera you will be using is a standard webcam. The program to run the
camera, BTVpro, has the ability to give you a relatively clear picture
of an object
moving in front of the camera.
You will utilize a piece of graph paper that is pasted to a vertical board as your
background. The dark and light lines on the graph paper will help you see where
your object is in each frame.
The object you wi
ll utilizing will be a small steel sphere.
By comparing one frame to the next, you will be able to track the path of the steel
sphere as a function of time. From this path, you can then determine the velocities
and the accelerations of the sphere as it pr
ogresses through the experiment.
1.4
Procedure
Open the BTVpro program. This program will run the webcam that you will take
your data with.
Check the settings of the webcam.
o
Under the “Video Display” set the camera to
640x480
The Kinematics of Free Fall
1

4
o
Under the “Setting” menu
Go to “
Video Device Settings
”
then
choose the “Adjustments” tab
and make sure your settings for color and image match.
Make
sure you hit the “tab” button right before you click “OK” to make
sure the settings are inputted.
You might need to reduce the Exposur
e time.
Now click the “Preferences” menu
Select the “Movie Capture” and check the settings.
The Kinematics of Free Fall
1

5
Record the frame rate, f
frame
rate
, of the webcam.
Set up the camera facing the vertical grid board (Figure 1

4). Check to see
that the grid board is as larg
e as possible on the computer screen; however
make sure that the entire grid can be seen.
Figure 1

4
1.
Now you are ready to collect data
One person will drop the steel ball within 3cm of the vertical grid shortly after
the other person clicks on the
“captu
re movie” button
.
Let the steel ball bounce once off the lab desk.
When the steel ball hits the lab desk the after the bounce stop the movie capture.
A new window on the computer should show up (this is the video you just
recorded).
Now you have your fir
st set of data
The Kinematics of Free Fall
1

6
Use the forward and backward buttons on the lower right side of the new
window to see that you have at least
EIGHT
frames that have the ball visible
and inside the grid. These eight frames should be for the bounce only (
not
the time from wh
en the ball was dropped to when it bounced the first time).
If you have less than eight “good” frames redo the run.
If you have at least eight “good” frames then start at the first “good” frame
and copy the location of the steel ball onto your graph paper
.
o
Try to mark only the center of the steel ball. If the ball appears to be
blurry, mark the center of the blur.
Number the points on your data sheet frame by frame starting with “1”.
Next repeat the experiment with one small change.
Instead of dropping t
he ball, slightly bounce the steel ball horizontally across
the vertical grid.
o
The ball should bounce the first time near one of the lower corners of
the grid and hit the table again near the other lower corner.
Again make sure that at least
eight
frames a
re “good”.
Again copy the steel ball position onto your data sheet.
o
Make sure you include both the horizontal (x) and vertical (y)
coordinates.
Once you have both the dropped and bounced data sets close out BTVpro and do not save
anything.
1.5
Interpretation
of Measurements
1.
Using a ruler, the frame rate and your raw data construct a table of time,
t
j
, and
vertical position,
y
j
, for the first data set. Construct your table using Excel (see the
tutorial starting from page 11 in Appendix A). Plot
y
j
versus
t
j
as
shown in the Excel
tutorial and find the acceleration using the equation of the Excel calculated
polynomial trend

line.
a
y
vs t
.
2.
Use the data on
y
j
and
t
j
to construct a table of instantaneous midpoint velocities,
v
m,j
,
of the steel ball as a function of
time,
t
m,j
. Use this data to construct the average
acceleration,
j
a
, during each time interval,
,,1
m j m j
t t
. Finally, average the
individual average accelerations,
j
a
, to get an combined a
verage acceleration,
a
table
,
for the run.
3.
Using Excel, plot the midpoint velocity,
v
m,j
, versus midpoint time,
t
m,j
. Using the
equation of the Excel calculated linear trend

line, find the acceleration,
a
v
vs
t
, (the
slope), find the velocity,
v
i
, the ste
el ball had when it first crossed the bottom grid
The Kinematics of Free Fall
1

7
line. Next, find the time,
t
max
, when the maximum height was reached. Using your
fitted line for the position vs. time graph, calculate the maximum height the steel ball
reached.
4.
Repeat only step 1 for t
he second set of data but construct both the vertical (
y
j
) as
well as the horizontal (
x
j
) directions.
The Kinematics of Free Fall
1

8
The Kinematics of Free Fall
1

9
Data Set 1: Vertical Drop
Only
The Kinematics of Free Fall
1

10
Data Set 2: Vertical and Horizontal Drop
The Kinematics of Free Fall
1

11
PHY 102M

Lab Report

Experiment 1: The Kinematics
of Free Fall
NAME
CLASS TIME
2.
Experimental Results
Frame Rate: f
frame
rate
=
Data Set 1:
j
t
j
(s)
y
j
(m)
1
2
3
4
5
6
7
8
9
10
11
12
j
t
m,j
(s)
v
m,j
(m/s)
j
a
(m/s
2
)
1

2
3
4
5
6
7
8
9
10
11
t
max
=
v
i
=
H
max
=
a
table
=
a
y
vs
.
t
=
a
v
vs
.
t
=
Don't forget to attach your plots of vertical position versus time and vertical velocity
versus midtime.
The Kinematics of Free Fall
1

12
PHY 102M

Lab Report

Experi
ment 1: The Kinematics of Free Fall
Data Set 2:
j
t
j
(s)
x
j
(m)
y
j
(m)
1
2
3
4
5
6
7
8
9
10
11
12
v
i x
=
a
x vs
.
t
=
v
i y
=
a
y
vs
.
t
=
Don't forget to attach
your
plots of horizontal posit
ion versus time and vertical position
versus time.
3. Useful Formulas
framerate
j
f
j
t
1
,
,
2
j j
m j j
t t
t t
1
,
1
,
j j
m j
j j
x x
v v
t t
j
m
j
m
j
m
j
m
j
t
t
v
v
a
,
1
,
,
1
,
1
C
Bt
t
A
H
max
2
max
max
)
(
where A, B, and C are from your gra
ph.
4. Conclusion Section
This section contains answers to questions about this lab provided by your lab
instructor.
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο