Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 1 - The Two-Dimensional Motion of a Projectile Equipment: air table strobe (furnished by instructor) meter sticks Each group will be given a strobe picture of a two-dimensional motion that simulates the motion of a projectile. By “projectile” it is meant any object thrown or dropped, which accelerates under the influence of the force of gravity. If the object is given an initial horizontal component of velocity it will execute a two-dimensional motion. That is, its path through space will require a specification of two numbers at each point (vertical and horizontal coordinate) for a complete description. The motions we have been studying so far , in contrast, have been one-dimensional in that, to describe the trajectory only one position coordinate need be specified at each point. All motions take place in three dimensional space, of course, but two and one-dimensional motions are confined to planes and lines respectively. The motion depicted is only a simulation of a true projectile as it represents the motion of a puck sliding over the surface of a tilted air table. The puck is projected up the incline at an angle so that its velocity has both horizontal and vertical velocity components. A true projectile is in vertical free fall. The motion studied here will differ from a true projectile in one respect: the vertical acceleration will be less than g . However, the essential features of the air table motion are representative of what one would observe for a true projectile. Velocity Components An object executing a two dimensional motion as described above, will have two position variables that change with time. That is, at each point along its trajectory the object will have (in general) different vertical and horizontal coordinates. At every instant we can then define the rate at which the vertical (or y) coordinate is changing and call this the vertical (or y) velocity, and we can define a corresponding quantity for the horizontal (or x) coordinate. We call this second quantity the horizontal (or x) velocity. These quantities are also referred to as components of the overall velocity vector. Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 2 - One way to picture a motion in two dimensions is to think of it as two, one-dimensional motions. In the case of a projectile, that would mean, one vertical and the other horizontal. Each coordinate, vertical and horizontal, traces out a trajectory (position versus time). The horizontal trajectory can be thought of as the shadow of the object, as it moves, projected onto the ground. The vertical trajectory is a shadow projected onto a wall. The respective velocity components represent the velocities of the respective shadows. We can think along similar lines as we seek to describe the motion of the air table puck replacing vertical with the “up the inc ...

1. Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 1 -
The Two-Dimensional Motion of a Projectile
Equipment: air table strobe (furnished by instructor)
meter sticks
Each group will be given a strobe picture of a two-
dimensional motion that simulates
the motion of a projectile. By “projectile” it is meant any object
thrown or dropped,
which accelerates under the influence of the force of gravity. If
the object is given an
initial horizontal component of velocity it will execute a two-
dimensional motion. That
is, its path through space will require a specification of two
numbers at each point
(vertical and horizontal coordinate) for a complete description.
The motions we have
been studying so far , in contrast, have been one-dimensional in
that, to describe the
trajectory only one position coordinate need be specified at each
point. All
motions take place in three dimensional space, of course, but
two and one-dimensional
motions are confined to planes and lines respectively.
The motion depicted is only a simulation of a true projectile

2. as it represents the motion
of a puck sliding over the surface of a tilted air table. The puck
is projected up the incline
at an angle so that its velocity has both horizontal and vertical
velocity components. A
true projectile is in vertical free fall. The motion studied here
will differ from a true
projectile in one respect: the vertical acceleration will be less
than g . However, the
essential features of the air table motion are representative of
what one would observe for
a true projectile.
Velocity Components
An object executing a two dimensional motion as described
above, will have two
position variables that change with time. That is, at each point
along its trajectory the
object will have (in general) different vertical and horizontal
coordinates. At every instant
we can then define the rate at which the vertical (or y)
coordinate is changing and call
this the vertical (or y) velocity, and we can define a
corresponding quantity for the
horizontal (or x) coordinate. We call this second quantity the
horizontal (or x) velocity.
These quantities are also referred to as components of the
overall velocity vector.
Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 2 -

3. One way to picture a motion in two dimensions is to think of
it as two, one-dimensional
motions. In the case of a projectile, that would mean, one
vertical and the other
horizontal. Each coordinate, vertical and horizontal, traces out a
trajectory (position
versus time). The horizontal trajectory can be thought of as the
shadow of the object, as it
moves, projected onto the ground. The vertical trajectory is a
shadow projected onto a
wall. The respective velocity components represent the
velocities of the respective
shadows. We can think along similar lines as we seek to
describe the motion of the air
table puck replacing vertical with the “up the incline” and
horizontal with this direction’s
perpendicular counterpart. In the diagram above the lengths
�
Δx and
�
Δy represent the
horizontal and vertical displacements of the puck over an
interval that represents some
small fraction of the entire motion. In the strobe photo all of the
time intervals are of the
same duration. (Usually 0.1 or 0.2 s. Check with your
instructor). If we know the value of
the time interval (we will call it
�
Δt) then the quantities

4. �
Δx Δt and
�
Δy Δt should
represent horizontal and vertical velocity values. Technically
these are the average
velocities over the interval, but since the interval is of
relatively short duration, they
approximate the instantaneous values at the beginning of the
interval. So we can write
�
Vx ≈
Δx
Δt
�
Vy ≈
Δy
Δt
for the instantaneous velocity components at each point in time.
Note that the variable t
here grows in increments of magnitude Δt. That is, t=0, Δt, 2Δt,
3Δt etc..

5. Producing a Strobe
Note: The risk of electrical shock while using the air table is
significant. All strobes
should be produced either by the instructor or with direct
supervision of students by the
instructor.
Elevate the air table at one end by using a book or another
object that will elevate that
end of the table. The table should be elevated at roughly a 10°
angle. Set the strobe rate
(10 per second is recommended) and place the foot switch on
the floor. Make sure that
the air supply tubes are connected to both pucks. There is a
metal chain within each tube
that makes electrical contact with the puck. Make sure that
neither of these is dangling
out of the tube. Place a piece of strobe paper under the pucks.
Plug in and/or turn on both
�
Vy
�
Vx
�
V
�
Δx !�
Δy !

6. Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 3 -
the air supply pump and the strobe spark generator. Do not step
on the foot switch yet.
Place one puck in the far corner of the table and test to see if
the other puck will slide
freely on the table. You want to produce a parabolic motion by
giving the puck a push
that yields both initial horizontal and vertical velocity
components. Try a few practice
runs first without stepping on the foot switch. When you
actually take data it is best to
start the motion first and then almost immediately after the puck
begins its motion
stepping on the foot switch. This technique eliminates the initial
acceleration produced by
the push from the data. Finally, produce a motion with the
switch engaged.
Data Analysis: Phase 1
On the strobe picture that your group is given, carefully
sketch lines that represent the
horizontal and vertical displacements of the object over all
intervals displayed. Note that
near the ends of the trajectory there may be erratic points that
don’t fit the overall pattern.
Ignore these points in your analysis. They are usually caused by
a collision between the
puck and the boundaries of the table over which it glides and/or
represent the puck being
pushed to initiate its motion. We are interested in the “free’”
motion of the puck after it

7. has been released and before it collides with any physical
boundaries. After the lines have
been drawn, measure their length (in cm) and enter those values
in the table below. For
each value also add either a “+” or a ”-”sign. A “+” sign should
accompany all left to
right horizontal displacements and all upward vertical
displacements. A “-” sign should
accompany right to left and downward displacements.
Note that each measurement should be associated with a time
value (t not Δt). We will
think of that time value as being the clock reading at the
beginning of each interval. Let
the first point that you include in your analysis represent t=0.
Then the second point
represents t=Δt, the third t=2Δt etc.. Then the first set of
displacements occur at t=0, the
second at t=Δt etc.. Once you’ve measured and entered the
displacements, calculate the
horizontal and vertical velocities (include + and - signs) as
described in the previous
section. Enter those values into the table as well. When you
have completed this process
plot the following graphs using the Graphical Analysis
software.
Vx versus time
Vy versus time

8. Δx
Δy
Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 4 -
Your instructor will show you how if you are not familiar with
the software. Your
instructor will also show you how to “fit” your data to straight
line (linear) functions1.
Label your axes, title your graphs and print each graph.
Attached the printed graphs to
your report.
The slope of each line (obtained from the best fit equation)
represents the average value
of the quantity ΔV/Δt. So, the slope of the Vx versus time graph
is the horizontal
acceleration, or ax, and the slope of the Vy versus time graph is
the vertical acceleration,
or ay. Determine the slopes as described above.
ax=________ (include units: cm/s2 or m/s2)
ay=________ (include units: cm/s2 or m/s2)
Question
Look at your velocity-time graphs. Does either graph (or both)
describe a velocity that is constant in time? Explain your
answer.

9. Question
Are the horizontal and vertical accelerations about the same size
or is one much larger
than the other?
Question
Consider the vertical motion of the puck. Look at your Vy
versus time graph. Is the puck
slowing down along the upward path at the same rate that it
speeds up on the downward
path? i.e. Is the acceleration uniform (to within reasonable
error)?
Question
In which direction is the vertical acceleration directed (up or
down the incline) during the
upward portion of the puck’s motion? In which direction is the
vertical acceleration
directed during the downward portion?
Data Analysis: Phase 2
Either using your strobe or your previous data table, fill in the
table below with position
(y and x) and time values (t). Use a coordinate system where the
origin, (x,y)=(0,0), is at
the position of the first dot in your strobe pattern. Note that the
previous table contains

10. 1 Section 2.1 in the Introductory Materials generally describes
the process of least
squares fitting of functions to data.
Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 5 -
displacements not positions. You can, however, use this data to
generate positions a given
set of displacements Δy0,Δy1,Δy2,...and Δx0,Δx1,Δx2,... can be
converted into positions
y0,y1,y2,,... and x0,x1,x2,...using the following procedure. First
set values for y0 and
x0 as these are the initial position coordinates. They are the
coordinates when t=0.0 s. For
the coordinate system we are using both y0 and x0 are equal to
0.0 cm. Then, the next set
of position values, y1 and x1, can be calculated using the
formulas y1=y0+Δy0 and
x1=x0+Δx0. These are the coordinates at t=Δt. The coordinates
at t=2Δt can be calculated
using the formulas y2=y1+Δy1 and x2=x1+Δx1. This process
can be continued for the rest of
the data set. Alternately, the position values can be measured
directly from the strobe
pattern. Once you have collected data, use Graphical Analysis
to plot graphs of y vs. t
and x vs .t. Using the software fit the y vs. t data to a quadratic
function and the x vs. t
graph to both linear and a quadratic functions. Print out all
three graphs (2 of x vs. t) and
attach them to your report.

11. The fit to the y vs. t data is of the form:
y=____ + _______ t +_______ t2
Fill in the blanks above based on your best fit equation
including correct units for all
quantities. Recall that a position-time function in this form
describes motion at constant
acceleration2.
The quadratic fit to the x vs. t data is of the same form.
x=____ + _______ t +_______ t2
Fill in the blanks in this equation based on your best fit
equation including correct units
for each quantity.
The linear fit to the x vs. t data is an equation of the form
x= _____ + ______ t
Fill in the blanks in this equation including correct units for
each quantity. Recall that a
position-time function in this form describes a motion at
constant velocity.
Question
The quadratic equations discussed above are of the form
�
y = y0 + vy0t +
1
2
ayt

12. 2and
�
x = x0 + vxot +
1
2
axt
2The quantities with “0” subscripts are initial (t=0) values. The
accelerations are equivalent to the accelerations previously
discussed. . Use your best fit
results (above) to determine ay and ax and compare these
results to the previously
determined ax and ay.
2 See section 1.5 in the Introductory Materials. This is an
example of comparing data to
an idealized model to help interpret the trend.
Two Dimensional Motion…, copyright Doug Bradley-Hutchison
page - 6 -
Question
The linear best fit equation generated from the x vs. t data is
consistent with a constant
velocity motion. It is valid if the acceleration ax is zero or very

13. small. How does ax
compare to ay? Does the linear function appear to accurately
describe the x vs. t data? If
so, how is the accuracy of this fit related to the relative size of
ax?