PHYS 31, SCU Physics Dept.

, Spring 2024 Name:

Lab Section (day and time) Lab Partners:

Lab 3: Projectile Motion

Introduction: Today you will investigate projectile motion for an object in free fall - that is, an object launched
into the air that falls back to the ground under the influence of gravity only. To describe the projectile’s motion you
will use a coordinate system where the positive y-axis is vertically upward and the positive x-axis is horizontal and
in the direction of the initial launch. To simplify the analysis today, you may assume the gravitational acceleration
(g = 9.8 m/s2 ) is constant, so that ax = 0 and ay = −g, and you may ignore air resistance.
The equations of motion in the x and y directions for a projectile launched with an initial velocity v0 (vector!)
at an angle θ are given by (ignoring air resistance):

vx (t) = vx0 = constant (1)

x(t) = x0 + vx0 t (2)

vy (t) = vy0 − gt (3)

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y(t) = y0 + vy0 t − gt2 (4)
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where t is the time. The x and y components of the initial velocity v0 are vx0 = v0 cos(θ) and vy0 = v0 sin(θ).
Remember that vx = vx0 because there is no acceleration in the x direction.
Consider a projectile launched from an initial height H above the ground with an initial velocity v0 at an angle
θ with respect to the horizontal, as shown in Figure 1. Using Eqs. 1-4 you should be able to show (optional!) that
the projectile’s full path (vertical location y as a function of horizontal location x) can be written as:

gx2
y = H + tan(θ)x − (5)
2(v0 cos(θ))2

Notice that Eq. 5 is just the equation of the parabola shown in Figure 1! Last but not least, notice that the total
horizontal distance traveled by a projectile in time t is its range, R:

R = vx0 t = v0 cos(θ)t (6)

Getting Started: The experimental set-up follows the geometry shown in Figure 1. A steel ball (projectile) will
be launched using a mini projectile launcher. The spring launcher can be pulled to three different position settings;
today you will always release the steel ball from the second position. A protractor on the side of the launcher will
allow you to set the desired launch angle, ranging from 0◦ to 90◦ . Carbon paper taped to the floor will be used to
mark where the projectile lands, allowing you to measure the horizontal distance traveled. With this equipment, you
will determine the initial launch velocity and find which launching angle gives the longest range.
You must wear safety glasses (provided) in the lab today!

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carbon paper

Figure 1: A projectile is launched at an angle θ off a platform with initial height H above the around. The projectile
lands the on carbon paper, a distance R away from the base of the platform. (We chose to ignore the finite length
of the launcher muzzle here because it is small compared to the total distance traveled by the projectile.)

Part 1: Finding the initial launch velocity

Here you will use two different methods to determine the initial velocity of a steel ball when released from the
second launch setting of your projectile launcher (the second “click”).

1. Orient the launcher at 45◦ . Use the meter stick and plumb-bob provided to measure the height H of the muzzle
above the floor. H = ±
2. Fire away! Make sure you hold the launcher firmly to avoid recoil effects. Perform a couple of practice shots
to locate where the ball lands and align the carbon paper with the trajectory. Then take real data, where you
record for each trial the time-of-flight t, range R, firing angle θ and launch height H. Record your measurements
in a table. Include an estimate of the uncertainty associated with each measurement and be sure all column
headings include units. Take at least 3 data sets (more, if the data is not consistent) to reduce the overall
uncertainty in your final, averaged data.

3. Average the values in each of your raw data columns and record the results here:
Ravg = , tavg = ,

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4. Getting the initial speed v0 from range measurements: Use Eq. 5 to solve for the launch speed v0 as
a function H, θ and range, R. Then use your averaged data to determine v0 of your projectile. Include units!
v0 =

5. Getting the initial speed v0 from time: Use Eq. 6 and your averaged data to determine the initial launch
speed v0 using the measured flight time t, θ and R. v0 =

6. Compare your two results for v0 . Based on your experimental methods, which launch speed value do you think
is more accurate? What were the biggest sources of error in your measurements?

7. Use either of your results for the launch speed v0 to calculate the initial velocity of your projectile.
v0 (vector!)=

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Part 2: Launch angle and maximum range:

1. If the projectile in your earlier experiments had been launched from ground level, the launch angle of 45◦ would
have provided the largest possible range. P redict if the launch angle corresponding to maximum range for your
geometry (H>0) should be larger or smaller than 45◦ . State your reasoning.

2. Experimentally determine the launch angle that gives you maximum range. Using the same launch
setting on the projectile gun as before (position 2), measure the horizontal range of your projectile as a function
of launch angle for θ = 0◦ to 80◦ in 10 degree increments. To get consistent data, fire at each launch angle at
least twice and average your range values. Make sure you have the same height, H, every time you change the
launch angle. Make a table of your experimental data.

3. Plot your Range data! Plot the measured Range (y-axis) vs. θ (x-axis). Include x- and y- error bars on
each data point that show your approximate uncertainty in the measured values. Draw a smooth curve through
your data (including the error bars). From your plot, identify the critical angle that gives the largest range.
Was your earlier prediction correct?