Air Resistance Report
Group 10: Swayam Lotake, Rafay Zaheer, Brianna Bonnila
Abstract
Our goal of this experiment is to explore the influence of density and shape on the free
fall motion of objects through a simulation developed in Excel. By varying the cross-sectional
areas and densities of different objects, we aim to analyze their respective impacts on the time
taken for the objects to reach the ground from a given initial height and velocity.
The simulation's validity is confirmed through manual calculations for comparison. In
this lab, 10 different materials with varying degrees of density were tested to determine the
magnitude of the terminal velocities.
will go further into the analysis of the properties in a different section of this report. What
was concluded was that the lower the density, the slower the terminal velocities and the higher
the faster the terminal velocities. Materials like aluminum and gold showed very different results
from what we estimated what would happen.
Introduction
This air resistance lab consists of creating a simulation in Excel that will measure how
density and the shape of an object will play a role with how long it takes for a falling object to
reach the ground given an initial height and initial velocity. In this experiment, we took different
objects with different cross-sectional areas to analyze the differences they had with each other.
We also did the same with the density of the object to see what trends were occurring. To ensure
our simulation was working, we did our own calculation by comparing results. This is different
from our other air-resistance trials because we are now working with the mechanics of airresistance and analyzing the extent of its impact. In our first simulation we wanted to create a
simple model that included the weight of the object as the only force acting on the object. In our
second simulation, we included the function of air-resistance, which like friction, opposes the
free fall motion of the object. In our final lab, the question was, how do factors within the
function of air-resistances have effects on the free fall motion of the object.
Theory
This air-resistance lab consisted of three main activities; the first activity involved
setting up a simple model of an object in free fall. Our goal was to mimic a
scenario of an object that is falling and predict how long it will take to land,
consider multiple start heights and initial velocity. Our second main activity
involved the same model but now it will include air resistance which is determined
by how fast the object is moving. The slower the object is moving the lesser the
degree of the resisitance on the object. The faster the object travels, the greater the
force of air-resistance. Air-resistance is based off the density of the object, the
cross-sectional area of the object, and the drag coefficient of air-resistance. The
same way contact friction has a coefficient of friction. In our third main activity,
we wanted to know how the motion of the object would be affected by tweaks in
the interworking of air-resistance. We could do this by changing the density of the
object by using other materials or change the cross-sectional area of the object by
changing the shape of it.
Methods
This experiment was a simulation with three parts and many activities. As it was a
simulation, a computer and the access to excel was a must. Part one was to create a simulation
that tracked the path of an object that was thrown with various masses, initial velocities, and
initial heights without air resistance. This is done by finding the acceleration in the y axis and
then calculating the new height every delta t seconds. Using this information, a Height vs Time
graph was created. Part two was analyzed in the same manner but air resistance was included.
Lastly, for part three, we were instructed to find the time to hit the ground, maximum height
achieved and the terminal velocity of a sphere of 0.05-meter radius with ten different densities.
Data and Analysis
In this part one, we simulated the behavior of an object under the sole influence of
gravity, using the equations vf = vi + a(t) and xf = xi + v(t). We determined the object's final
velocity at each interval and tracked its changing position over time. The resulting graph shows a
decrease in height over time, indicating acceleration due to gravity. The varying slopes of the
curve suggest the presence of gravity, with negative slopes indicating downward acceleration.
Below is the data for this part.
For part two, the same experiment as part one was carried out but while including the
force due to drag - air resistance. Below are the preset value used and the data acquired.
trial
radius ball
density ball
area
0.05
2700
0.00785375
density air
drag coeff
1.25
0.5
b
0.002454297
Where F_drag = bV^2
opposite the motion
1
2
3
4
5
height(m)
20
50
25
10
31.415
time(y=0)
3.3-3.35
4.45-4.5
3.5-3.55
2.8-2.85
3.8-3.85
The figure above displays the preset values we used for air resistance and the five trial
simulations that we conducted. For this part of the air resistance experiment, the ball radius was
set to 0.05 meters and was assumed to be an aluminum ball with a density of 2700 kilogram per
meter cubed. The density of air was set to 1.25 kilogram per meter cubed and the drag coefficient
was 0.5. There should be no differences between the simulated results and the analytical results
in Figure 1.
The graph above displays an optional graph created to be a useful comparison between
the path of an object with and without air resistance. The first graph shows the position vs time
graph of an object with a mass of 1.413171669 kilograms thrown from a height of 100 meters
and an initial velocity of 25 meters per second upwards without any air resistance. While the
second graph shows the same scenario but with air resistance. The bottom graph uses the
aluminum ball with the density and other values as displayed in the table on the precious page.
From the simulations, we can obtain that without air resistance, the object reaches a maximum
height of 132.51 meters, while without air resistance, it reached 130.83 meters.
For part three, the same experiment as part two was carried out but three different values
were measured – maximum height reached with an initial velocity of 20 meters per second,
terminal velocities and time to hit the ground. These values were measured for ten different balls
of the same radius but different materials and densities. Belowe is the data from the simulation.
Trial
Density
1
2
Time
4.652700 4.7
4.558930 4.6
Velocity
Height
Material
20.196 75.13188 Aluminum
20.689 136.6369 copper
3
4
5
6
7
8
9
10
4.5519302 4.6
4.557870 4.7
4.751550 4.8
4.65765 4.65
4.64507 4.66
4.72200 4.75
4.5510490 4.6
4.5513310 4.7
20.807 200.8832 gold
20.659 128.2714 Iron
19.707 56.92568 Calcium
20.569 109.7848 Tin
20.475 97.07025 titanium
20.044 67.81937 salt
20.721 148.0915 silver
20.761 166.8136 Hafnium
In the above data, it can be noticed that for many materials, the time is the same. This can
be explained by the fact that the conducted simulations measured valued every delta t seconds, in
the cases for this experiment, the delta t was set to 5 seconds.
Discussions & Error Sources
This simulation allows us the understanding of what happens to the variables of a
falling object. When we analyzed the data the simulation gave us, we were able to see the
velocity, mass, and height or position of the ball at a specific time. When doing this we can
see the effects that air resistance or lack of has on the ball’s mass and velocity. For example,
when air resistance was not included in the simulation, its acceleration remained constant,
but when we included it, we were able to see how the acceleration changed. We compared
how different materials’ density acted in air resistance vs. no air resistance without ever
throwing an object. This simulation can help us understand what happens to an object when
variables are changed. You can compare how does the acceleration or velocity differs if the
object had a different mass.
Considering the structure of this lab there are no error sources that can influence or affect
results. Since this is a simulation, it consists of plugging in values in the correct spot in the
formula. However, human error, such as typos, assumptions, or estimate errors can still occur so
to combat this we double checked it by doing calculations by hand to make sure the simulation
was running in the correct direction.
Conclusion
We used our excel simulation to track the path of an object to find how density and shape
affected the time it takes for an object to reach the ground after being thrown up into the air. We
also measured the path with and without air resistance. By running this experiment, we were able
to prove that density does affect the time it takes to reach the ground, find its maximum height,
and its terminal velocity.