Projectile Motion and the Range Equation
You have learned that the motion of any object moving through the air
affected only by gravity is an example of projectile motion. Examples of
projectile motion include a basketball thrown toward a hoop, a car driven off a
cliff by a stunt person, and a marble launched from the CPO marble launcher.
Projectile motion is also called two-dimensional motion because it depends
on two components: vertical and horizontal. In this Investigation, you
will determine a mathematical model (the range equation) that predicts the
range of the marble given launch angle and initial velocity.
Analyzing the motion of the marble in two dimensions
How can you predict the range of the marble? Since gravity pulls down and not sideways, the motion of the
marble must be separated into components. It makes sense to pick one component (y) in the vertical
direction aligned with gravity. The other component (x) is then chosen to be in the horizontal direction,
perpendicular to the force of gravity. The diagram below shows the velocity of the marble (v) at three
points in its trajectory, resolved into x and y components, vx and vy
a. Use the diagram above to explain why projectiles travel in a curved path called a trajectory.
b. How does the marble's velocity in x change over the time of the flight? How does its velocity my
change over the time of flight?
' Understanding the velocity equations
The object of this investigation is to find and test a model that will predict the range of the marble from the
initial velocity and launch angle—the range equation.
a. The first step is to separate the velocity of the marble into x and
y components. Use the triangle formed by velocities (at right) to
express vx and vy in terms of the initial velocity, v, and the sine vy
and cosine of the launch angle, θ.
When the initial velocity is separated into x and y components, vx
Equations la-2b give the relationships between the motion variables separately for x and y. In these
equations the subscript 0 refers to the initial values at launch. Equations la and 1b are for the marble's
velocity while equations 2a and 2b are for the marble's position.
b. Since gravity does not pull up or sideways, one of the accelerations (ax, ay) is -g, and the other is zero.
Rewrite Equations la-2b leaving out terms that are zero and substituting your previous results for v0x
c. The purpose of this exercise is to find a theory that predicts where the marble will land (the range, ∆x)
given the initial velocity and launch angle. This problem can be solved in several steps. First, assume
that gravity acts only on the y component of velocity. Solve for the time it takes the marble to reach its
maximum height (where vv = 0).
d. Since gravity does not pull sideways, the x component of the marble's velocity is not affected and
remains constant. Use Equation 2a to calculate the range (∆x) from v0x and the total time of flight (this
will be the range equation).
Improving the range equation
The range equation did not work very well at small angles or low initial velocities. This difficulty
occurs because the range is measured on the floor even though the marble is actually launched at in
initial height yi above the floor, as shown in the drawing below. As a result, the marble travels an extra
distance xc before it reaches the floor. This additional distance is not accounted for by the previous range
Old range ∆x
Understanding quadratic equations
Start with the equation:
a. Write the new equation. In this form, it is a quadratic equation.
b. Quadratic equations have the form ax2 + bx + c = 0, where a, b, and c are constants and x is a
variable. Your equation is a quadratic equation for t, the time of flight for the marble. Take your
equation and group the terms so that they look like a quadratic equation for t. Solve for solutions
that y = 0, when the marble hits the ground.
c. Solve the equation for t using the quadratic formula. One of the solutions will be positive,
corresponding to the marble hitting the ground in front of the marble launcher. The other
solution will be negative. What does this solution correspond to? Use the positive solution and
derive an equation for the range by substituting your result for t into ∆x = vt cosθ. Your result
should only depend on the initial velocity v and the launch angel θ.
Testing the new theory
a. Launch the marble at 8 different angles (from low to high) using the same spring setting
each time. Record your data. Then use your launch information to calculate the
theoretical range using the original range equation and again with the improved range
Launc Tim Initial Measure Old New
h eA Velocit d range Theoretica Theoretica
Angle y l Range l Range
b. Make a graph of the range vs. angle for the experimental data, the old theory, and the new
c. How does your new theory compare with your measurements? Your answer should
include a numerical comparison based on calculating the percent difference between
theory and experiment. For example, suppose for one angle and speed the old theory
predicted 1.5 meters, the new theory predicted 2.5 meters, and the measurement was 2.35
meters. The measurement is 6% different from the new theory but 57% different from the
old theory. The fact that the old theory showed a deviation of 57% and the new theory
shows a deviation of only 6% means that the new theory is a better theory because it is
closer to the observations.
Question: How repeat-able are experiments with the marble launcher?
In this Investigation, you will determine the repeatability of marble launcher experiments.
You have probably discovered that the marble does not land in exactly the same place every time. This is true
even when the initial velocity and launch angle are the same for consecutive launches. A characteristic that
is shared by all real experiments is that it is generally not possible to make two measurements that are
exactly identical. In this Investigation, you will determine the repeatability of range measurements with the
1. A piece of graph paper will allow you to mark the landing points for each launch. Tape the graph paper
down to the floor.
2. Use a strip of masking tape on the floor to make sure that the marble launcher is set back in the
same place every time.
Distance (m) Measured
Doing the experiment pre-measured each shot
1. For the experiment, you need to make ten identical launches with the same spring setting and
launch angle, and record exactly where the marbles land. The best way to do this is to set up a
graph paper target area as shown above. Every time a marble lands, it will make a dent in the
paper. Mark each dot and give it a trial number.
2. Measure and record the launch angle, time from photogate A, initial velocity, and the distance from
the launcher to the target.
3. Use the timer to make sure your 10 launches have as close to the same initial velocity as you can
measure. This means you will have to mark (and ignore) launches that show different times from
photogate A. You might have to make 20 launches to get 10 that have the same initial velocity.
Keep track of which trials are the launches you will use and which ones you will ignore.
4. Try to be very consistent in how you release the launch lever. '
5. Ranges of a few meters work best for this experiment. The larger the range, the larger your paper
will have to be to catch all the landings. An 8 1/2 x 11 piece of graph paper will work up to a few
meters, but you will need a larger sheet to measure the repeatability of seven meter launches.
Analyzing the data
1. Use a ruler to determine the x and y coordinates of the ten good landings. Record the coordinates.
2. At the bottom of your data table, show the average of the x and y coordinates. These coordinates
correspond to the average range. Plot the average range on your target sheet with a cross and circle
as shown in the example.
3. The deviation (R), in the far right column of the table , is the distance from the average range to each
landing point. You may measure this with a ruler or calculate it using the formula for the distance
between two points on a plane. Measure and record the deviation for each landing point in the table.
4. Calculate the average deviation for all landings and record it at the bottom of the last row. Draw a
circle on your target sheet with the center at the average range position with a radius equal to the
Launch # X Y R
Accuracy and precision
The word accuracy describes how close an experiment comes to measuring the ‘true’ value.
Since all measurements are subject to some amount of variability (or error) a discussion of
accuracy must include an estimate for the variability in the measurement.
a. How accurate was your measurement of the range of the marble launcher? Your answer should
include a quantitative discussion of the error in terms of the deviations you measured.
The word precision describes how closely spaced a group of measurements are. A very precise
experiment can produce measurements that do not differ by very much from on trial to the next.
For example, a meter stick with ruling one millimeter apart is precise to one millimeter. For a
measurement of 1 meter this precision is 0.1%.
b. How precise is the marble launcher experiment? Express your answer as a percent of the total
What does your experiment say about predicting the outcome of another marble launch
under the same conditions? Scientific predictions about the future are made in terms of
probability and chance. In the example 6 of 10 launches fell within a 4 centimeter circle. If the
ten launches were a good representation of the performance of the marble launcher then it
would be reasonable to infer that 6 of the next 10 launces would also fall within this circle.
It can also be predicted that there is a 60% chance (6 out of 10) that nay single launch will fall
within the 4 cm circle. In statistical terms we could say that this experiment produced a value of
3.06 plus or minus 0.04 meters for the range of the marble. This result also implies that there is
a 60% probability that the next launch will be within the interval 3.06 plus or minus 0.04 m.
The 8 centimeter interval (plus or minus 0.04) represents the 60% confidence interval for this
measurement. The accepted standard for confidence level in the scientific community is 65%.
a. Use your data to estimate the radius for a 50% probability for a similar launch to land inside the
circle. Express your measured range as a value plus or minus this radius.
The two theories about the range of the marble produce predictions that are different. The
experiment must be sensitive enough to be able to tell which theory is more correct. If the
error associated with the measurement is larger than the difference between the theories, the
experiment is inconclusive.
b. Does your experiment disprove the old range equation? Your answer should account for your
determination of the errors in the experiment.
c. Does the experiment agree or disagree with the new theory you derived earlier? TO answer this
question, indicate whether there is a reasonable chance that any observed differences are due to
random fluctuations or represent real (reproducible) effects.