Experiment #A6: Projectile Motion –
Kinematics in 2-D
Dr. Anne Caraley
(revised F12, F14; rewritten F17; revised and renumbered F19)
(revised for Distance Learning alternative for Fall 2020)
Recommended Reading
Halliday/Resnick/Walker, Chapter 4 or Cutnell/Johnson/Young/Stadler, Chapter 3
OpenStax “College Physics” https:
openstax.org/details
ooks/college-physics, Chapter 3
PHY XXXXXXXXXXLab Manual Experiment #A5: Free Fall Motion – Kinematics of 1-D Motion
Other Useful References
Hyperphysics http:
hyperphysics.phy-astr.gsu.edu/hbase/traj.html (projectile motion trajectories)
I. Introduction and Objectives
The understanding of the motion of a projectile in two dimensions is fundamental to ou
understanding of how gravity behaves, baseball, and, unfortunately, also to human warfare.
Medieval military engineers had all manner of fanciful formulas for predicting where a
cannonball would land. It was Galileo who realized that such projectile motion could be
separated into a horizontal component, independent of gravity, and a vertical component
dependent on gravity. In the ideal case, projectile motion assumes the role of air resistance to be
negligible, as with Free Fall motion. Of course, in the real world that is not always appropriate
or even useful. For example, in sports, air resistance often plays an important role – the curve
all in baseball, the top spin lob in tennis, and the slice (or hook) drive in golf, to name a few.
In today’s experiment, a projectile (a small ball) will be launched from a table top, striking the
floor some distance away. An experimental determination of the ball’s flight time will be based
on a measurement of the ball’s launch angle and its horizontal displacement at the end of its
trajectory. Knowledge of the ball’s initial velocity, launch angle, and its height above the landing
point will also allow for a theoretical prediction of the flight time of the ball, for comparison.
For two studies (with different launch angles), a steel ball will be used so that the influence of ai
esistance on the ball will be minimized. A third study may be conducted with a plastic ball to
observe both qualitatively and quantitatively the role of air resistance.
Student’s should view today’s experiment as the culmination of their lab-related activities of the
past few weeks – in the techniques employed in the collection and analysis of their experimental
data, and in their preparation of the required full Formal Report. All measurements should be
made keeping in mind the precision of the instrument used and thus an uncertainty recorded.
(Measurements will be made both by hand and with a computer.) Detailed diagrams of the
apparatus will be required as part of the Formal Report so students should include them in thei
notebooks as part of a program of careful record keeping the day the experiment is conducted.
The data analysis should be conducted keeping in mind, and taking into account, both
instrumental and statistical uncertainties. Propagation of those uncertainties will be necessary.
Students will also be required to use a spreadsheet program to support conclusions drawn from
their experimental results with appropriate theoretical predictions and calculations, including
ealistic graphs (simulations) of projectile trajectories. A Formal Report is required, so students
should plan their time accordingly – before, during, and after the experiment.
Projectile: Page 1 of 19
https:
openstax.org/details
ooks/college-physics
http:
hyperphysics.phy-astr.gsu.edu/hbase/traj.html
II. General Background –
The Kinematic Equations in Two Dimensions
To begin our study of Projectile Motion, we must recall the fundamentals of Free Fall Motion.
Specifically, we must remember that the term “free fall” is used to describe the ideal (air-
esistance free) motion of an object under the influence of gravity alone – whether the object is
dropped from rest, thrown upward, or thrown downward. Recall also that the use of the term
“free fall” applies to one-dimensional motion where the initial velocity is confined to the vertical
dimension. Projectile motion is just an expansion of these principles to two-dimensions.
Of course, in the real world, we observe projectiles to be traveling in three-dimensions, but it
turns out that a projectile’s ideal path (where the effects of air resistance and/or the object’s spin
are negligible), its trajectory can be judged to be confined to one plane. (Think about it; what
would you perceive if you were traveling on the projectile itself?) For mathematical simplicity,
we can therefore always let the observed motion define the x-y plane! See Figure 1. As we have
seen previously, the choice of origin for such an x-y plane will be dictated by the particulars of a
specific scenario. Logic should dictate that the choice simplifies the problem.
Figure 1: Sketch of a Projectile Motion trajectory for an object launched from an initial
position higher than its landing position. Also shown here are an x-y coordinate system (origin
and orientation), the horizontal displacement, initial velocity, and launch angle. (D. Zych)
The first step in an expansion to two dimensions is to define separate variables and equations
pertinent to each of the dimensions. (Yup! Two sets of everything.) This separation is allowed
ecause the motion in the horizontal direction is completely independent of the motion in the
vertical direction – a consequence of the mutual orthogonality of the axes of the rectilinear x-y-z
coordinate system. Following the equations presented for the Free Fall experiment, the
kinematic equations in two dimensions can be written as:
Xf = Xo + Vox t + 1/2 ax t2 and Yf = Yo + Voy t + 1/2 ay t2, Equation 1a
o
ΔX = Xf – Xo = Vox t + 1/2 ax t2 and ΔY = Yf – Yo = Voy t + 1/2 ay t2, Equation 1
and
Vfx = Vox + ax t and Vfy = Voy + ay t, Equation 2
and
Vfx2 = Vox2 + 2 ax ΔX and Vfy2 = Voy2 + 2 ay ΔY, Equation 3
where the x-related and y-related variables are associated with the horizontal and vertical
directions, respectively. What is key to realize about two-dimensional motion is that although the
motions (the x and y components of the position, the x and y components of velocity, and x and y
components of acceleration) are separate, the time clock (t) is the same for the two dimensions.
The common time, t, is what connects the otherwise independent motions.
Projectile: Page 2 of 19
Depending on the scenario, the knowledge of an object’s motion in one of the two dimensions
will often dictate the overall behavior of the object. For scenarios involving projectile motion,
the governing dimension is the vertical one because of the role of the acceleration due to gravity.
In stark contrast, there is not any acceleration acting in the horizontal dimension at all, ax = 0.
Applying these conditions to the kinematic equations given previously, and by choosing the
downward direction to be taken as negative, ay = –g, the earlier equations simplify to
Xf = Xo + Vox t and Yf = Yo + Voy t – 1/2 g t2, Equation 4a
o
ΔX = Xf – Xo = Vox t and ΔY = Yf – Yo = Voy t – 1/2 g t2, Equation 4
and
Vfx = Vox and Vfy = Voy – g t, Equation 5
and
Vfx2 = Vox2 and Vfy2 = Voy2 – 2 g ΔY. Equation 6
Examination of these equations reveals immediately that the horizontal component of a
projectile’s velocity remains unchanged throughout its flight, (Equation 5 for Vfx). In othe
words, its horizontal position is directly proportional to the elapsed time, t. Consequently, the
measurement of a projectile’s final horizontal displacement can provide an experimental value of
its total time-of-flight, provided the object’s initial velocity and launch angle are measured as
well. This conclusion follows from Equation 4b for the horizontal dimension:
XXXXXXXXXXt = Δ X
V ox
= Δ X
V ocos (θ)
. Equation 7
A more subtle understanding gained from these equations is that total time-of-flight of the
projectile is ultimately governed by its motion in the vertical dimension (Equation 4a for Yf) –
ecause eventually, gravity
ings the object down. Consequently, a “theoretical” time-of-flight
of a projectile can be predicted, provided its initial velocity, launch angle, and starting and
ending elevations are known according to
t =
−(V osin (θ)) ± √(V o sin(θ))2 − 4 (−1/2g)(−ΔY )
2(−1 /2 g)
, Equation 8
where the quadratic equation has been applied Equation 4a for the vertical dimension. The
derivation follows:
Yf = Yo + Voy t – 1/2gt2
0 = (Yo-Yf) + Vo sin(θ)t – 1/2gt2
XXXXXXXXXX = ( -ΔY) + Vo sin(θ)t – 1/2gt2,
which when compared to the form of the quadratic equation
0 = C +Bt + At2
allows for the identification of the parameters
A = – ½ g, XXXXXXXXXXB = + Vo(sinθ), C = -ΔY,
and then the use of the standard form for the two roots of a quadratic equation,
t =
−(B) ± √(B)2 − 4 ( A)(C)
2(A)
. Equation 9
Projectile: Page 3 of 19
III. Building A Theoretical Model –
Flight Times for Trajectories of Projectiles
To understand the physical significance of the two solutions