Lab 5: Projectile Motion

Description
In this lab, you will examine the motion of a projectile as it free falls through the air. This
will involve looking at motion under constant velocity, as well as motion under constant acceleration.

Equipment
• Clear Plastic Hose

• Tape (optional)

• Ring Stand and Clamps

• Paper (optional)

• Meter Stick

• Carbon Paper (optional)

• Stopwatch

Introduction
In the study of mechanics, one deals with the kinematics of a body or the dynamics of a body.
Kinematics generally concerns the position, velocity, and acceleration of the body, while dynamics
concerns the forces acting on the body. They are connected by Newton’s Second Law that relates
the forces on a body to the induced acceleration by

F = ma (1)

Today we will examine projectile motion which, in it’s simplest form is the motion of an object
under constant acceleration. In our case this well be under the influence of gravity alone. In this
experiment we will observe projectile motion and gain practice in applying kinematic equations to
a typical problem encountered in physics, engineering, security, and aviation-related studies. Ex-
amples of projectiles include bullets, Intercontinental Ballistic Missiles, unguided (gravity) bombs,
baseballs, and airplanes in certain (mostly highly undesirable) situations.

Kinematics of Projectile Motion

In this laboratory experiment, our projectile consists of a metal ball bearing. We will observe
two distinct dynamical cases in this experiment, namely

1. Constant Velocity

2. Constant Acceleration (free fall)

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 We will start with Part 1 where the acceleration is zero, and therefore the velocity is constant.
Denoting the horizontal direction as x and applying the kinematic equations (a detailed explanation
of this can be found in Appendix D) we get
x = x0 + v0,x t (2)
vx = v0,x (3)
where all variables are defined as in the Appendix D.
In this context, we consider the forces in the horizontal direction (rolling friction, air drag) to
be negligible, thus, the horizontal accelerations are negligible, so that the horizontal velocity is
constant, as indicated in Equation 2. The position of the ball increases linearly in time.
Next, in Case 2, we will expand our experiment to include a constant acceleration. In free fall,
the only non-negligible force acting on the projectile is gravity, which we will consider constant in
the lab. If we retain our definition of x as being in the horizontal direction and define y to be in
the vertical direction, then this case’s kinematic equations are:
x = x0 + v0,x t (4)
vx = v0,x (5)
1
y = y0 + gt2 (6)
2
vy = −gt (7)
Please note that these equations are specifically tailored to this lab’s experimental procedure
and do not represent the generic casesee Appendix D for a discussion on the generic kinematic
equations applicable to any problem.
We will be able to determine the initial position and velocity of our projectile by repeating the
first portion of the lab. We will then accelerate the projectile under the force of gravity and by
measuring how far it travels horizontally, we can the projectiles vertical acceleration, g.

Procedure
Part 1: Constant Velocity

1. Clamp the upper end of the tube at some convenient height, such as 20 cm, above the table
and allow the loose end to lay on the table. Record this initial height in your lab book.
2. Create a table in your lab book where you will record the change in distance, the time elapsed,
and the average velocity for 5 trials.
3. Carefully measure and record the distance between the lower end of the hose and the edge of
the table top. This will be your change in distance, x − x0 .
4. You will propel the ball by dropping it through the tube. To keep the velocity as similar as
possible between trials, you may want to develop a system to keep your method of dropping
the ball consistent. Drop the ball several to practice measuring the time it takes to traverse
the table top from the end of the hose to the end of the table (your change distance). Then,
launch the ball and record the time it takes in your lab book. Repeat this for a total of 5
trials.

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Figure 1: Apparatus setup. The ring stand is used to hold the clear plastic hose; the hose is used
to launch the ball at a fairly repeatable speed. Note that you can place one coordinate frame on
the desk or table top, and another on the floor.

5. Calculate the velocity v0,x across the table top. Record these values in your table.

6. Carefully measure and record the height of the table H and record this value in Table 1.

Part 2: Constant Acceleration

1. Very accurately record the height of your table, y0 . What will be your final height, y?

2. Create a new table in which you will record the initial height of the tube, h; the change in
distance, x − x0 ; the time elapsed while rolling, tr ; and the range, R. You will also want
columns to record the initial velocity, vx ; the time of flight, tf ; and the calculated value of
gravity, g.

3. You will now repeat the measurement from Part 1, however instead of catching the ball after
it leaves the table, you will allow it to free fall to the ground and measure how far it goes.
First, try a practice run to see approximately where the ball hits the ground. Tape a piece
of carbon paper there so you can measure where it hit.

4. Run the experiment again, recording the distance the ball rolls across the table, the time it
takes to roll across the table (from these two values you can calculate the initial velocity),

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 and the range, which is the distance to moves before it hits the floor as measured from the
edge of the table.

5. Since you know the initial velocity and how far it moves horizontally (the range), you can
calculate the time of flight of the ball. Record this in your table.

6. Using the height of the table as your initial height, y0 , the ground as zero height, y = 0, and
your time of flight, t, calculate the gravitational constant. Check that your value is reasonably
correct before continuing.

7. Next, repeat this part 5 more times for different tube heights. If you have extra time, take
more measurements, this will improve your results.

Analysis

1. Calculate your average gravitational constant from Part 2.

2. We can also use graphs to determine relationships between values. Solving Equations 4 and
6 for g, we get
2yv 2
g= (8)
R)2
thus if you plot 2yv 2 on the vertical axis and (x − x0 )2 on the horizontal axis, the slope of
the best fit line should equal your value of g. Do this and compare the slope to the value
calculated in Question 1.

3. What possible sources of random error did you observe in this experiment? List as many as
you can think of?

4. What possible sources of systematic error, bias, or calibration error, if any, did you observe
in this experiment? List as many as you can think of.

5. What was the fractional error of your estimated value for the acceleration due to gravity?
Allowing for some random error, is your estimate an acceptable value for g?

6. Based on your results, do you consider the kinematic equations and our assumptions to be
validated by this experiment?

7. How did the average value of g compare with the individual table values of g? Find the
standard deviation and comment. Did this variation affect your best fit line in your graph?

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 Appendix D: A Detailed Description of Kinematic
Equations
From everyday experience, recall that the average velocity may be defined as the distance traveled
divided by how long it took to traverse that distance, or:
x − x0 ∆x
v̄ = = (1)
t − t0 ∆t
where positions are denoted by x and times by t; the subscript 0 indicates an initial quantity
and the lack of a subscript indicates a final quantity. Thus, this object is at position x0 at time
t0 , and at position x at time t. The Greek upper case letter ∆, ”Delta”, indicates the difference
between the final and initial quantities, or the change in that quantity. This equation may be
rearranged to yield:
x − x0 = v̄ · (t − t0 ) (2)
We may further rearrange Equation 2 to produce:

x = x0 + v̄ · (t − t0 ) (3)

Often, the initial time t0 is numerically 0, so the term t0 disappears, as in Equation 1.

Using the same logic, the average acceleration can be defined as:
v − v0 ∆v
ā = = (4)
t − t0 ∆t
which can be rearranged (with t0 = 0) to yield:

v = v0 + ā · t (5)

Special Case: Constant Acceleration

In this case, the average acceleration is just the singular, constant acceleration a (you can prove
this to yourself by computing the average of the numbers 3, 3, and 3 it’s just 3).
Consider how the velocity changes over a time interval. At time t0 , the velocity is v; after some
elapsed time t, the velocity v is given by Equation 5. The average velocity over this time interval
can be computed like any average of two numbers:
1 1 1 1
v= · (v0 + v) = · (v0 + (v0 + at)) = · (2v0 + at) = v0 + at (6)
2 2 2 2
Substituting this result into either Equation 3 or 1, we obtain the generic equation for position:
1
x = x0 + v0 t + at2 (7)
2
Equations 4 and 7 are the fundamental equations used in this experiment and are applicable to
any case involving a constant acceleration.

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 Figure 1: Visualizing kinematics.

General Solution Strategies

Often, it is easy to lose track of what’s what in all these equations. An easy way to visualize
what is happening is to consider a time interval defined as the time between t0 and t, as shown in
Figure 1.
Generally, you wish to determine x and v at time t. To do this, the general strategy is:

• Determine or identify the initial quantities x0 and v0 .

• Determine or identify the constant acceleration a.

• Determine or compute the elapsed time ∆t; if t0 = 0, then ∆t = t.

• Apply Equations 4 and 7 to compute x and v, respectively.

It is important to recall that v0 and a can be negative or positive (positions and times may be
as well, but this is a relatively rarer condition in most word problems or practical applications).
Positive values of v0 and a arise when their corresponding vector quantities point towards +∞ on
the directional axis (usually the x axis for one-dimensional motion) whereas negative values arise
when their corresponding vectors point towards −∞.

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