Modelling the motion of a falling styrofoam cup

Sreenidhi International School Physics Extended Essay

Candiadate Name: Baig Mirza Amanullah Candidate number: 004976-0013

1

MODELLING THE

MOTION OF A FALLING

STYROFOAM CUP

Subject : Physics.

Topic: Mechanics.

Candidate name: Baig Mirza Amanullah

Candidate number: 004976-0013

Supervisor name: Gyaneshwaran Gomathinayagam.

School: Sreenidhi International School.

Word count:



2

Abstract

The falling motion of a Styrofoam cup was modelled using a simulation which I programmed in

Vpython1.

An empty Styrofoam cup was dropped and its motion was compared with the predicted motion of

three models. The first model ignored the effects of friction and considered only free fall motion.

The second model added the effects of laminar drag force along with gravitational force. The third

model added the effects of turbulent drag force along with gravitational force. The second model

consisting of laminar drag force and gravitational force matched the actual motion of the

Styrofoam cup best and the predicted terminal velocity matched very closely with the actual

terminal velocity of the Styrofoam cup. The laminar drag coefficient b of the model was

0.00325 kg and the predicted terminal velocity of 1.22 ms-1

matched the terminal velocity value of

1.22 ms-1

measured using TRACKER analysis. The model was also used to make predictions

about the motion of the Styrofoam cup if it had been thrown down with non-zero velocities or if it

had some mass placed inside it.

(Word Count 199)

1 Scherer, David. "Vpython 3D Programming for Ordinary Mortals." Vpython. National Science Foundation,

1 Jan. 2000. Web. 13 Mar. 2015. <http://vpython.org/>.



3

Table of Contents Abstract ........................................................................................................................................................... 2

List of Figures .................................................................................................................................................. 4

List of Tables.................................................................................................................................................... 5

Acknowledgement .......................................................................................................................................... 6

Introduction .................................................................................................................................................... 7

Investigation ................................................................................................................................................... 7

Vpython program to model the motion of the Styrofoam cup ………………………………………………………9

Discussion of Results..................................................................................................................................... 15

Comparision of position vs time for actual and predicted data……………………………………………………15

Predicted drag force and net force vs time………………………………………………………………………………….16

Predicted velocity vs time for different initial velocities (under drag force)…………………………………17

Predicted velocity vs time for different masses kept inside Styrofoam cup (with drag force)………18

Conclusion ..................................................................................................................................................... 19

Evaluation ..................................................................................................................................................... 19

Bibliography .................................................................................................................................................. 20

Appendix ....................................................................................................................................................... 20

Raw Data .................................................................................................................................................... 20



4

List of Figures Figure 1: Screenshot of TRACKER software being used to track the position of the styrofoam cup while

falling ............................................................................................................................................................... 8

Figure 2: Comparison of position vs time of actual and predicted data ....................................................... 15

Figure 3: Predicted drag force and net force vs time .................................................................................... 16

Figure 4: predicted velocity vs time graph for different initial velocities (considering drag force) .............. 17

Figure 5: Predicted velocity vs time graphs for different masses kept inside the styrofoam cup (considering

drag force) ..................................................................................................................................................... 18



5

List of Tables Table 1 Position and velocity data of Styrofoam cup obtained from TRACKER analysis of the video of the

falling cup. ..................................................................................................................................................... 20

Table 2 Position and velocity data of Styrofoam cup obtained from Vpython simulation of the falling

Styrofoam cup for the model with drag force and gravitational field. ......................................................... 24



6

Acknowledgement

I am very grateful to my supervisor Mr. Gyaneshwaran G for his guidance in completing the

extended essay.

I would like to thank my physics teacher Mr. Thavamani T for his guidance.



7

Introduction

Every evening our school provides snacks in Styrofoam cups. I felt it would be interesting to

model its falling motion when it is dropped from a height. Styrofoam cup is very light and so will

experience the effects of air resistance significantly.

Air resistance is typically considered to be of two types – laminar drag and turbulent drag. In the

real world, for low speeds, laminar drag dominates, and for high speeds, turbulent drag dominates,

but mostly it is a mix of both. Thus, the best model should include the effects of both drag forces

in the model. However, I’ve only tried to use one of the drag forces at a time and tried to find out

which drag force fits the actual motion of the falling Styrofoam cup best.

The magnitude of turbulent drag force is given by the equation �� = −�|�|� where b is called the

drag coefficient. Thus, turbulent drag force is proportional to the square of the magnitude of the

velocity of the object.

The magnitude of laminar drag force is given by the equation � = −�|�| where b is again the

drag coefficient. Here, laminar drag force is proportional to the magnitude of the velocity of the

object.

Investigation

I took a video of the falling motion of a Styrofoam cup by dropping it from a height. I took the

usual precautions like keeping the camera surface parallel to the plane of the motion of the

Styrofoam cup, and ensuring that there was no side wind. I used an iphone to take the video in

slow motion mode. Slow motion mode lets us take videos in more frames per second. So we can

capture the position data at shorter time intervals.



8

After taking the video, I opened the video file in TRACKER software2.

Figure 1: Screenshot of TRACKER software being used to track the position of the styrofoam cup while falling

After loading the video in TRACKER, the first step is to choose the starting and ending frame of

the relevant motion to be tracked. Then using the meter scale in the background, I calibrated the

video using the calibration stick. Then I set the coordinate system with the origin at the point of

release of the Styrofoam cup. So the Styrofoam cup falls along the negative y axis after it is

dropped.

Then I created a ‘point mass’ to track the bottom-center of the Styrofoam cup’s position through

each frame by SHIFT+Click again and again, until I reached the last frame of the selected portion

2 Brown, Douglas. "Tracker Video Analysis and Modeling Tool for Physics Education."Tracker

Video Analysis and Modeling Tool for Physics Education. Open Source Physics, 1 Jan. 2015. Web.

13 Jan. 2015.



9

of the video which I decided to track. The y position, and y components of velocity and

acceleration are automatically generated in the table and can be copy pasted onto a spreadsheet.

The graph which I got for the position of the Styrofoam cup can be seen in the above screenshot.

Vpython program to model the motion of the Styrofoam cup

I wrote the following program in Vpython to model the motion of the Styrofoam cup with gravity,

with gravity and laminar drag, and with gravity and turbulent drag.

# STARTUP (Don't edit, typically)

from __future__ import division, print_function

from visual import *

from physutil import *

import csv

# VISUALIZATION & GRAPH INITIALIZATION

# ===========================================

# Setup window for plotting

pos_graph = PhysGraph(1)

vel_graph = PhysGraph(1)

# Setup Display window for visualization

scene = display(width = 700, height = 700, background = color.white, range = 10, title

= "Styrofoam cup falling model")

scene.autoscale = 0

#scene.center = (0,-6,0)

scene.center = (0,0,0)

# Create object for visualization

cup = sphere(color = color.blue, radius = 0.3, pos = (0,15,0))

# Create a track behind object to visualize trajectory

trail = curve(color = color.green, radius = 0.06)



10

# Create small sphere to mark the origin in Display window

origin = sphere(pos=vector(0,0,0), color = color.black, radius = 0.1)

# Create arrows to represent vector quantities in the Display window

DragForce_arrow = arrow(pos=cup.pos, axis = (0,0,0), color = color.black);

Weight_arrow = arrow(pos=cup.pos, axis = (0,0,0), color = color.red);

# SYSTEM PROPERTIES and INITIAL CONDITIONS

# ===========================================

# System Mass

#EDIT THIS (next one line):

cup.m = 0.0005

#cup.m = 0.0005+0.0041 # mass of crocodile clip = 4.1+-0.2 g

#Initial Conditions

#EDIT THIS (next three lines, as necessary)

cup.pos = vector(0,0,0)

cup.vel = vector(0,0,0)

#cup.vel = vector(0,-1.4,0) #setting non-zero downward initial velocity

#cup.vel = vector(0,1,0) # setting non-zero upward initial velocity of 1 m/s

#cup.vel = vector(0,10,0) # setting non-zero upward initial velocity of 10 m/s

t = 0 #the time when we choose to start our clock

# OTHER INITI#ALIZATION

# ===========================================



11

#

#Timestep

#EDIT THIS (next one line, as necessary)

#deltat = 0.001

deltat=0.008

#Interaction Initialization

#unit vector of velocity

vhat=cup.vel.norm()

#proportionality constant, b, for magnitude of drag force

b=0.00325

#b=0

#proportionality constant, g, for magnitude of weight force

#set g to magnitude of gravitational field near earth's surface

g=9.8

#g=3.6314 modified g based on acceleration observed in experimental data

#unit vector for y axis---choose "up" as direction of +y axis

jhat=vector(0,1,0)

#constant weight force acting downwards on styrofoam cup

w=-cup.m*g*jhat

Fd=vector(0,0,0)

Fnet=vector(0,0,0)

#OPTIONAL: Output model predictions to file (.csv format)

#To use, uncomment next five lines (delete leftmost # ONLY)

outputfile = open('Lab2_StyrofoamCupFalling_v01.csv', 'w') # Set name of output file.



12

# # NOTE: if the file already exists,

# # it will be overwritten

DataWriter = csv.writer(outputfile, delimiter=',', lineterminator='\n',quotechar='|',

quoting=csv.QUOTE_MINIMAL) # Create writer object

DataWriter.writerow(['Time (s)','Position (m)','Velocity (m/s)','Drag force (N)','Net

force (N)']) # Write column headers for time, position, and velocity

DataWriter.writerow([t,cup.pos.y,cup.vel.y,Fd.y,Fnet.y])

#print(t,cup.pos.y)

#Update the object's track

trail.append(pos = cup.pos)

#Plot something as a function of time


pos_graph.plot(t,cup.pos.y)

vel_graph.plot(t,cup.vel.y)

# CALCULATION LOOP(Motion Prediction and Visualization)

# ===========================================

while t < 1.58125:

#turbulent drag force proportional to square of velocity and directed opposite to

it

Fd = -b*cup.vel.mag2*vhat

print ("Fd=",Fd,"w=",w,"vhat=",vhat)

#Calculate Net Force

#EDIT THIS (next one line; add more lines if necessary)

Fnet = w + Fd

#Predict new velocity (Insert Newton's 2nd Law here)

#EDIT THIS (next one line)

cup.vel = cup.vel + (Fnet/cup.m)*deltat

vhat=cup.vel.norm()

#Predict new position



13

#EDIT THIS (next one line)

cup.pos = cup.pos + cup.vel*deltat

# advance the clock

t = t + deltat

print(t,cup.pos.y)

DataWriter.writerow([t,cup.pos.y,cup.vel.y,Fd.y,Fnet.y])

#Update the object's track

trail.append(pos = cup.pos)

#EDIT THIS: Draw arrows to show quantities of interest

DragForce_arrow.pos = cup.pos;

Weight_arrow.pos = cup.pos;

# determines how long to draw arrows to represent vectors; here, the value of

arrowscale tells us what arrow length (in meters) corresponds to a force with a

magnitude of 1 Newton

Force_arrowscale = 2.0/w.mag # max distance available on screen = max[(max(X)-

min(X)),(max(Y),min(Y))] = 1.34e+12 m

# max magnitude of force vector that needs to be

plotted = max(Fnet,Fnetpar,Fnetperp) = 2.79e+25

# scaling factor for the arrows of all force vectors

(all similar quantities should

# be plotted using same scaling factor so that we can

compare their magnitudes by

# observing their relative lengths) = max distance

available on screen/max magnitude of force vector

# So I obtained arrowscale1 = 1.34e+12/2.79e+25 =

4.80e-14, which is PERFEECCCCT!

DragForce_arrow.axis = Fd*Force_arrowscale;

Weight_arrow.axis = w*Force_arrowscale;

#Plot something as a function of time


pos_graph.plot(t,cup.pos.y)



14

vel_graph.plot(t,cup.vel.y)

#To speed up or slow down program execution

#Edit next line (larger number, faster execution)

rate(20)

#OPTIONAL: Output model predictions to file (.csv format)

#To use, uncomment next line

outputfile.close()

The part of the code which has the important physics equations is made bold.

The logic of the program is as follows:

The program assigns a sphere to represent the Styrofoam cup. So, the mass of the Styrofoam cup

is assigned to this sphere. The motion of this sphere will represent the motion of the Styrofoam

cup. Its initial conditions are set as zero velocity and its initial position is set as origin. The

acceleration value due to gravity, g is set as 9.8 ms-2

.

A loop condition is started and made to last for the duration of the video motion which we tracked.

The time of the last frame in our video is 1.58125 s. So we’ll run the loop by incrementing the

time in small intervals of deltat=0.008 s, till the final time reaches 1.58125 s. Then the loop ends

and the program stops.

During this loop, before each time the time value is incremented by deltat, the position and

velocity values are updated based on the net force equations and kinematics equations.

For example, let the initial velocity be v. After the time interval of deltat, the new velocity will be

the old velocity v + a*deltat, where a is the acceleration due to the net force acting on the object.

We know that =��

� and �� = � + �� where w = weight force and �� = drag force.



15

The expressions for weight and drag force are typed in the program. Thus, the new velocity value

is used to update the new position value using new x = v*deltat. Thus, the program keeps updating

the position and velocity of the Styrofoam cup based on the expressions we have typed for weight

force w and drag force Fd.

The correct value of the drag coefficient b is arrived at by trial and error. I ran the program and

generated the data of the velocity as a function of time. The program will output any data we want

as a .csv file which can be opened as a spreadsheet in EXCEL. By comparing this data with the

actual TRACKER data of the Styrofoam cup, I adjusted the value of b till both the data matched.

That’s how I figured out that the drag coefficient must be b = 0.00325.

Discussion of Results

The position and velocity data for the Styrofoam cup obtained from TRACKER analysis of the

video and generated by the Vpython program are in the Appendix.

Figure 2: Comparison of position vs time of actual and predicted data

-14

-12

-10

-8

-6

-4

-2

0

0 0.5 1 1.5 2

y /

m

t / s

yvst

ExperimentalDa

tag=9.8only

Predicted

b=0.00325



16

From the graph, it is clear that the model for drag force matches almost perfectly with the actual

data of position vs time of Styrofoam cup which was obtained from TRACKER analysis. If we

ignore the effects of air resistance, model predicts that the cup would have travelled a much larger

distance in the same time, as shown by the blue curve. The experimental data and the model with

drag force predicts that the slope of the position-time graph will become constant at around 0.5 s.

So that is when its velocity will become constant, in other words, it acquires terminal velocity.

Figure 3: Predicted drag force and net force vs time

From this graph, we can see that the drag force (shown in red) starts from zero, and quickly

reaches a plateau value, after which it remains constant. This means, the object has reached

terminal velocity. The weight force is constant throughout the motion of the object. So this plateau

is reached, when the object stops accelerating because the drag force has become equal to the

weight force in magnitude while being opposite in direction. Thus these two forces cancel each

other out, resulting in zero net force, which is seen by the blue curve. Just when the drag force

reaches its maximum value, the net force reaches zero value, after starting out with maximum

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Fo

rce

, N

time / s

Predicted Drag force (N) and Net force (N) vs

time

Fd

Fnet



17

magnitude negative value at the start of the motion. The maximum magnitude negative value of

the initial net force is just equal to the weight of the object.

Figure 4: predicted velocity vs time graph for different initial velocities (considering drag force)

This graph shows the predictions made by the program for the velocity of the Styrofoam cup as a

function of time if the Styrofoam cup were thrown with different initial velocities instead of just

dropping it from rest. To make these new predictions, I don’t have to write any new code in the

program, since the laws of motion and net force equation are still the same. I just have to change

the value of the initial velocity from zero to each of these values like -1 m/s, -1.4 m/s, -2 m/s and -

5 m/s.

We find that for all the cases, the Styrofoam cup is predicted to attain the SAME terminal velocity

of 1.22 ms-1

at nearly the same time of around 0.36 s! If a Styrofoam cup starts with very high

non-zero velocity, it takes the same time to attain terminal velocity as a Styrofoam cup which

starts out with a much lower initial velocity. The reason is that greater initial velocity results in

greater drag force, since turbulent drag force is proportional to square of the magnitude of the

-6

-5

-4

-3

-2

-1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

v m

/s

time t / s

predicted v vs t for different vi [with drag force]

v vs t for vi=-1 m/s [with drag

force]"

v vs t for vi=0 m/s [with drag

force]


force]


force]

v vs t for vi = -1.4 m/s [with

drag force]



18

velocity of the object. So the acceleration is greater and the greater change in velocity happens in

the same time and it attains terminal velocity around the same time.

It would be very interesting to try this out for real and verify these predictions.

Figure 5: Predicted velocity vs time graphs for different masses kept inside the styrofoam cup (considering drag force)

From this graph, we can see that the heavier Styrofoam cup attains terminal velocity much later

than the lighter Styrofoam cup. This makes sense, since the net force will become zero for heavier

cup only when the drag force becomes equal to the much larger weight of the heavier cup. The

drag force will attain this larger value only after attaining the greater velocity as shown by the blue

curve. The lighter cup requires a smaller drag force to cancel its weight, and hence smaller

velocities are sufficient to cause the drag force to cancel the weight, and make the Styrofoam cup

achieve terminal velocity.

It would be interesting to verify this graph by experimental data obtained from video analysis

using TRACKER.

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2

ve

rtic

al

ve

loci

ty,

v /

m/s

time, t / s

predicted v vs t for different masses kept inside

styrofoam cup [with drag force]

m=0.0046: v vs t; with drag

force; vi=0

v vs t; m=0.0005; vi=0; with

drag force



19

Conclusion

1. Vpython is an extremely simple and powerful 3D programming language which can be used to

simulate and model any physics phenomena. The program does the integration numerically to

calculate the new position and velocity based on the net force equation which we feed into the

program. Thus, all we need to know to simulate a real-life situation, is a knowledge of the net

force equation. Unknown constants can be found out by trial and error method by comparing the

output data of the program with the TRACKER data obtained from motion analysis of the videos.

2. The model made many interesting and logically sound predictions about the motion of the

Styrofoam cup in new situations like if it had different non-zero initial velocities and if it had

different initial masses.

Evaluation

1. I should have taken the videos and analysed them using TRACKER and compared the actual

data with the predicted data. That would have been ideal. Unfortunately, I ran out of time.

Tracking the data by clicking the position of the object in each frame is very time-consuming,

especially since the number of frames runs into thousands when we take slow-motion videos at

say, 240 frames per second. So there is tons of raw data to process.

2. The model should incorporate a mix of both types of drag forces to obtain an even better match

with the actual position vs time data of the Styrofoam cup.

3. The model’s predictions for different masses and different initial velocities of the Styrofoam

cup should be verified with actual data.



20

Bibliography Brown, Douglas. "Tracker Video Analysis and Modeling Tool for Physics Education."Tracker Video Analysis

and Modeling Tool for Physics Education. Open Source Physics, 1 Jan. 2015. Web. 13 Jan. 2015.

Scherer, David. "Vpython 3D Programming for Ordinary Mortals." Vpython. National Science Foundation,

1 Jan. 2000. Web. 13 Mar. 2015. <http://vpython.org/>.

Appendix

Raw Data

Table 1 Position and velocity data of Styrofoam cup obtained from TRACKER analysis of the

video of the falling cup.

mass_A y coordinate of position y component of velocity y component of acceleration

time , t / s y / m v_{y} / ms-1 a_{y} / ms

-2

0 0

0.008333 -0.00111 -0.19994

0.016667 -0.00333 -0.13329 0

0.025 -0.00333 -0.13329 -9.14013

0.033333 -0.00555 -0.33323 -6.8551

0.041667 -0.00889 -0.26659 4.570063

0.05 -0.01 -0.19994 -2.28503

0.058333 -0.01222 -0.33323 -11.4252

0.066667 -0.01555 -0.39988 0

0.075 -0.01888 -0.33323 -7.02974

0.083333 -0.0211 -0.47849 -11.7162

0.09125 -0.02666 -0.6152 -2.34325

0.099583 -0.0311 -0.46653 7.02974

0.107917 -0.03443 -0.46653 -2.28503

0.11625 -0.03888 -0.53317 -9.14013

0.124583 -0.04332 -0.59982 -6.8551

0.132917 -0.04887 -0.66647 -9.14013

0.14125 -0.05443 -0.73311 11.42516

0.149583 -0.06109 -0.53317 -4.57006

0.157917 -0.06331 -0.66647 -4.57006

0.16625 -0.0722 -0.79976 -9.14013

0.174583 -0.07664 -0.66647 9.140127

0.182917 -0.08331 -0.73311 -6.8551

0.19125 -0.08886 -0.73311 -11.4252

0.199583 -0.09553 -0.93305 -9.14013

0.207917 -0.10441 -0.93305 18.28025

0.21625 -0.11108 -0.59982 -6.8551

0.224583 -0.11441 -0.93305 0

0.232917 -0.12663 -0.86641 6.855095



21

0.24125 -0.12885 -0.53317 9.140127

0.249583 -0.13552 -0.86641 -15.9952

0.257917 -0.14329 -0.79976 -4.57006

0.26625 -0.14884 -0.86641 -11.4252

0.274583 -0.15773 -1.06635 -36.5605

0.282917 -0.16662 -1.39958 2.285032

0.29125 -0.18106 -1.19964 27.42038

0.299583 -0.18661 -0.73311 29.70541

0.307917 -0.19328 -0.79976 -31.9904

0.31625 -0.19994 -1.19964 -9.14013

0.324583 -0.21327 -1.13299 11.42516

0.332917 -0.21882 -0.79976 -4.57006

0.34125 -0.2266 -1.26629 -15.9952

0.349583 -0.23993 -1.19964 0

0.357917 -0.24659 -1.06635 9.140127

0.36625 -0.2577 -1.19964 0

0.374583 -0.26659 -0.9997 -20.5653

0.382917 -0.27436 -1.46623 9.372987

0.39125 -0.29102 -1.16205 21.08922

0.399167 -0.29325 -0.75191 7.02974

0.4075 -0.30324 -1.26629 -35.1487

0.415833 -0.31435 -1.26629 -2.28503

0.424167 -0.32435 -1.26629 -2.28503

0.4325 -0.33546 -1.33294 0

0.440833 -0.34656 -1.26629 11.42516

0.449167 -0.35656 -1.13299 2.285032

0.4575 -0.36545 -1.19964 -4.57006

0.465833 -0.37655 -1.26629 25.13535

0.474167 -0.38655 -0.79976 -9.14013

0.4825 -0.38988 -1.19964 -27.4204

0.490833 -0.40655 -1.59952 -27.4204

0.499167 -0.41654 -1.39958 20.56529

0.5075 -0.42987 -1.39958 6.855095

0.515833 -0.43987 -1.19964 9.140127

0.524167 -0.44987 -1.26629 11.42516

0.5325 -0.46097 -1.06635 0

0.540833 -0.46764 -1.13299 -6.8551

0.549167 -0.47986 -1.33294 -13.7102

0.5575 -0.48985 -1.26629 -2.28503

0.565833 -0.50096 -1.39958 -6.8551

0.574167 -0.51318 -1.39958 2.285032

0.5825 -0.52429 -1.33294 27.42038

0.590833 -0.5354 -0.9997 11.42516

0.599167 -0.54095 -0.9997 2.285032

0.6075 -0.55206 -1.13299 -34.2755



22

0.615833 -0.55983 -1.39958 -6.8551

0.624167 -0.57538 -1.46623 13.71019

0.6325 -0.58427 -0.9997 20.56529

0.640833 -0.59205 -1.13299 -34.2755

0.649167 -0.60315 -1.59952 -18.2803

0.6575 -0.6187 -1.53288 20.56529

0.665833 -0.6287 -1.13299 25.13535

0.674167 -0.63759 -1.13299 -2.28503

0.6825 -0.64758 -1.19964 -9.14013

0.690833 -0.65758 -1.26629 2.285032

0.699167 -0.66869 -1.19964 -9.37299

0.7075 -0.67758 -1.36711 -9.37299

0.715417 -0.6909 -1.50382 0

0.72375 -0.70201 -1.26629 16.40273

0.732083 -0.71201 -1.19964 -4.57006

0.740417 -0.72201 -1.33294 0

0.74875 -0.73423 -1.26629 -2.28503

0.757083 -0.74311 -1.26629 -6.8551

0.765417 -0.75533 -1.46623 -4.57006

0.77375 -0.76755 -1.33294 18.28025

0.782083 -0.77755 -1.13299 2.285032

0.790417 -0.78643 -1.26629 -15.9952

0.79875 -0.79865 -1.46623 -18.2803

0.807083 -0.81087 -1.53288 -2.28503

0.815417 -0.8242 -1.53288 15.99522

0.82375 -0.83642 -1.26629 11.42516

0.832083 -0.8453 -1.26629 6.855095

0.840417 -0.85752 -1.26629 6.855095

0.84875 -0.86641 -1.06635 4.570063

0.857083 -0.87529 -1.19964 -36.5605

0.865417 -0.8864 -1.66617 -15.9952

0.87375 -0.90306 -1.59952 22.85032

0.882083 -0.91306 -1.13299 38.84554

0.890417 -0.92195 -0.9997 -2.28503

0.89875 -0.92972 -1.13299 -25.1353

0.907083 -0.94083 -1.46623 -4.57006

0.915417 -0.95416 -1.26629 6.855095

0.92375 -0.96193 -1.19964 -4.57006

0.932083 -0.97415 -1.46623 -9.14013

0.940417 -0.98637 -1.33294 13.71019

0.94875 -0.99637 -1.19964 0

0.957083 -1.00637 -1.33294 -27.4204

0.965417 -1.01858 -1.66617 -11.4252

0.97375 -1.03414 -1.59952 13.71019

0.982083 -1.04524 -1.33294 22.85032



23

0.990417 -1.05635 -1.26629 6.855095

0.99875 -1.06635 -1.19964 4.686493

1.007083 -1.07635 -1.2304 -9.37299

1.015 -1.08634 -1.36711 -23.4325

1.023333 -1.09856 -1.59952 -18.746

1.031667 -1.113 -1.66617 20.56529

1.04 -1.12633 -1.26629 25.13535

1.048333 -1.13411 -1.13299 -2.28503

1.056667 -1.14521 -1.39958 -11.4252

1.065 -1.15743 -1.33294 -9.14013

1.073333 -1.16743 -1.46623 4.570063

1.081667 -1.18187 -1.39958 -2.28503

1.09 -1.19076 -1.33294 9.140127

1.098333 -1.20408 -1.39958 -2.28503

1.106667 -1.21408 -1.26629 -6.8551

1.115 -1.22519 -1.53288 -6.8551

1.123333 -1.23963 -1.46623 22.85032

1.131667 -1.24963 -1.06635 9.140127

1.14 -1.2574 -1.26629 -20.5653

1.148333 -1.27073 -1.53288 -6.8551

1.156667 -1.28295 -1.33294 9.140127

1.165 -1.29295 -1.33294 4.570063

1.173333 -1.30517 -1.33294 -31.9904

1.181667 -1.31516 -1.73282 4.570063

1.19 -1.33405 -1.53288 29.70541

1.198333 -1.34071 -0.93305 22.85032

1.206667 -1.3496 -1.26629 -29.7054

1.215 -1.36182 -1.46623 0

1.223333 -1.37403 -1.26629 20.56529

1.231667 -1.38292 -1.06635 -13.7102

1.24 -1.39181 -1.46623 -18.2803

1.248333 -1.40736 -1.53288 2.285032

1.256667 -1.41735 -1.26629 15.99522

1.265 -1.42846 -1.33294 -4.57006

1.273333 -1.43957 -1.33294 -13.7102

1.281667 -1.45068 -1.53288 11.42516

1.29 -1.46512 -1.26629 20.56529

1.298333 -1.47178 -0.9997 2.285032

1.306667 -1.48178 -1.33294 -13.7102

1.315 -1.494 -1.26629 -7.02974

1.323333 -1.50288 -1.36711 0

1.33125 -1.51621 -1.43547 -7.02974

1.339583 -1.52621 -1.33294 28.11896

1.347917 -1.53843 -1.06635 0

1.35625 -1.54398 -1.13299 -20.5653



24

1.364583 -1.55731 -1.59952 -18.2803

1.372917 -1.57064 -1.39958 34.27548

1.38125 -1.58064 -0.9997 2.285032

1.389583 -1.5873 -1.26629 -38.8455

1.397917 -1.60174 -1.79946 -11.4252

1.40625 -1.61729 -1.46623 31.99044

1.414583 -1.62618 -1.13299 25.13535

1.422917 -1.63618 -1.13299 -11.4252

1.43125 -1.64506 -1.26629 -20.5653

1.439583 -1.65728 -1.53288 -11.4252

1.447917 -1.67061 -1.46623 13.71019

1.45625 -1.68172 -1.26629 -2.28503

1.464583 -1.69172 -1.46623 -9.14013

1.472917 -1.70616 -1.53288 6.855095

1.48125 -1.71726 -1.26629 11.42516

1.489583 -1.72726 -1.33294 13.71019

1.497917 -1.73948 -1.13299 25.13535

1.50625 -1.74615 -0.79976 0

1.514583 -1.75281 -1.13299 -20.5653

1.522917 -1.76503 -1.26629 -61.6959

1.53125 -1.77391 -1.93276 -11.4252

1.539583 -1.79724 -1.86611 50.2707

1.547917 -1.80502 -0.73311 70.83598

1.55625 -1.80946 -0.73311 -15.9952

1.564583 -1.81723 -1.06635 -27.4204

1.572917 -1.82723 -1.19964

1.58125 -1.83723

Table 2 Position and velocity data of Styrofoam cup obtained from Vpython simulation of the

falling Styrofoam cup for the model with drag force and gravitational field.

Time (s) Position (m) Velocity (m/s) Drag force (N) Net force (N)

0 0 0 0 0

0.008 -0.00063 -0.0784 0 -0.0049

0.016 -0.00188 -0.15648 2.00E-05 -0.00488

0.024 -0.00375 -0.23361 7.96E-05 -0.00482

0.032 -0.00622 -0.30917 0.000177 -0.00472

0.04 -0.00928 -0.3826 0.000311 -0.00459

0.048 -0.01291 -0.45339 0.000476 -0.00442

0.056 -0.01708 -0.5211 0.000668 -0.00423

0.064 -0.02176 -0.58538 0.000883 -0.00402

0.072 -0.02693 -0.64596 0.001114 -0.00379



25

0.08 -0.03255 -0.70266 0.001356 -0.00354

0.088 -0.03859 -0.75539 0.001605 -0.0033

0.096 -0.04503 -0.80412 0.001854 -0.00305

0.104 -0.05182 -0.84889 0.002101 -0.0028

0.112 -0.05894 -0.88982 0.002342 -0.00256

0.12 -0.06635 -0.92705 0.002573 -0.00233

0.128 -0.07404 -0.96076 0.002793 -0.00211

0.136 -0.08197 -0.99116 0.003 -0.0019

0.144 -0.09012 -1.01847 0.003193 -0.00171

0.152 -0.09846 -1.04294 0.003371 -0.00153

0.16 -0.10698 -1.06477 0.003535 -0.00136

0.168 -0.11565 -1.08422 0.003685 -0.00122

0.176 -0.12446 -1.10149 0.00382 -0.00108

0.184 -0.1334 -1.1168 0.003943 -0.00096

0.192 -0.14244 -1.13034 0.004054 -0.00085

0.2 -0.15158 -1.14231 0.004152 -0.00075

0.208 -0.1608 -1.15285 0.004241 -0.00066

0.216 -0.1701 -1.16214 0.004319 -0.00058

0.224 -0.17946 -1.17031 0.004389 -0.00051

0.232 -0.18888 -1.17749 0.004451 -0.00045

0.24 -0.19835 -1.18379 0.004506 -0.00039

0.248 -0.20787 -1.18932 0.004554 -0.00035

0.256 -0.21742 -1.19417 0.004597 -0.0003

0.264 -0.22701 -1.19841 0.004635 -0.00027

0.272 -0.23662 -1.20213 0.004668 -0.00023

0.28 -0.24627 -1.20539 0.004697 -0.0002

0.288 -0.25593 -1.20823 0.004722 -0.00018

0.296 -0.26562 -1.21072 0.004744 -0.00016

0.304 -0.27532 -1.2129 0.004764 -0.00014

0.312 -0.28504 -1.2148 0.004781 -0.00012

0.32 -0.29477 -1.21646 0.004796 -0.0001

0.328 -0.30451 -1.21791 0.004809 -9.07E-05

0.336 -0.31427 -1.21918 0.004821 -7.92E-05

0.344 -0.32403 -1.22029 0.004831 -6.92E-05

0.352 -0.3338 -1.22125 0.00484 -6.04E-05

0.36 -0.34358 -1.2221 0.004847 -5.27E-05

0.368 -0.35336 -1.22283 0.004854 -4.60E-05

0.376 -0.36315 -1.22348 0.00486 -4.02E-05

0.384 -0.37294 -1.22404 0.004865 -3.51E-05

0.392 -0.38274 -1.22453 0.004869 -3.06E-05

0.4 -0.39254 -1.22496 0.004873 -2.67E-05

0.408 -0.40234 -1.22533 0.004877 -2.33E-05

0.416 -0.41214 -1.22566 0.00488 -2.03E-05

0.424 -0.42195 -1.22594 0.004882 -1.78E-05

0.432 -0.43176 -1.22619 0.004885 -1.55E-05



26

0.44 -0.44157 -1.2264 0.004886 -1.35E-05

0.448 -0.45138 -1.22659 0.004888 -1.18E-05

0.456 -0.4612 -1.22676 0.00489 -1.03E-05

0.464 -0.47101 -1.2269 0.004891 -8.97E-06

0.472 -0.48083 -1.22703 0.004892 -7.83E-06

0.48 -0.49065 -1.22713 0.004893 -6.83E-06

0.488 -0.50046 -1.22723 0.004894 -5.96E-06

0.496 -0.51028 -1.22731 0.004895 -5.20E-06

0.504 -0.5201 -1.22739 0.004895 -4.53E-06

0.512 -0.52992 -1.22745 0.004896 -3.96E-06

0.52 -0.53974 -1.2275 0.004897 -3.45E-06

0.528 -0.54956 -1.22755 0.004897 -3.01E-06

0.536 -0.55938 -1.22759 0.004897 -2.63E-06

0.544 -0.5692 -1.22763 0.004898 -2.29E-06

0.552 -0.57903 -1.22766 0.004898 -2.00E-06

0.56 -0.58885 -1.22769 0.004898 -1.74E-06

0.568 -0.59867 -1.22772 0.004898 -1.52E-06

0.576 -0.60849 -1.22774 0.004899 -1.33E-06

0.584 -0.61831 -1.22775 0.004899 -1.16E-06

0.592 -0.62813 -1.22777 0.004899 -1.01E-06

0.6 -0.63796 -1.22779 0.004899 -8.80E-07

0.608 -0.64778 -1.2278 0.004899 -7.68E-07

0.616 -0.6576 -1.22781 0.004899 -6.70E-07

0.624 -0.66742 -1.22782 0.004899 -5.84E-07

0.632 -0.67725 -1.22783 0.004899 -5.10E-07

0.64 -0.68707 -1.22783 0.0049 -4.45E-07

0.648 -0.69689 -1.22784 0.0049 -3.88E-07

0.656 -0.70672 -1.22784 0.0049 -3.38E-07

0.664 -0.71654 -1.22785 0.0049 -2.95E-07

0.672 -0.72636 -1.22785 0.0049 -2.57E-07

0.68 -0.73618 -1.22786 0.0049 -2.25E-07

0.688 -0.74601 -1.22786 0.0049 -1.96E-07

0.696 -0.75583 -1.22786 0.0049 -1.71E-07

0.704 -0.76565 -1.22786 0.0049 -1.49E-07

0.712 -0.77548 -1.22787 0.0049 -1.30E-07

0.72 -0.7853 -1.22787 0.0049 -1.13E-07

0.728 -0.79512 -1.22787 0.0049 -9.89E-08

0.736 -0.80494 -1.22787 0.0049 -8.63E-08

0.744 -0.81477 -1.22787 0.0049 -7.53E-08

0.752 -0.82459 -1.22787 0.0049 -6.57E-08

0.76 -0.83441 -1.22787 0.0049 -5.73E-08

0.768 -0.84424 -1.22788 0.0049 -5.00E-08

0.776 -0.85406 -1.22788 0.0049 -4.36E-08

0.784 -0.86388 -1.22788 0.0049 -3.80E-08

0.792 -0.87371 -1.22788 0.0049 -3.32E-08



27

0.8 -0.88353 -1.22788 0.0049 -2.89E-08

0.808 -0.89335 -1.22788 0.0049 -2.52E-08

0.816 -0.90317 -1.22788 0.0049 -2.20E-08

0.824 -0.913 -1.22788 0.0049 -1.92E-08

0.832 -0.92282 -1.22788 0.0049 -1.68E-08

0.84 -0.93264 -1.22788 0.0049 -1.46E-08

0.848 -0.94247 -1.22788 0.0049 -1.27E-08

0.856 -0.95229 -1.22788 0.0049 -1.11E-08

0.864 -0.96211 -1.22788 0.0049 -9.70E-09

0.872 -0.97194 -1.22788 0.0049 -8.46E-09

0.88 -0.98176 -1.22788 0.0049 -7.38E-09

0.888 -0.99158 -1.22788 0.0049 -6.44E-09

0.896 -1.0014 -1.22788 0.0049 -5.62E-09

0.904 -1.01123 -1.22788 0.0049 -4.90E-09

0.912 -1.02105 -1.22788 0.0049 -4.27E-09

0.92 -1.03087 -1.22788 0.0049 -3.73E-09

0.928 -1.0407 -1.22788 0.0049 -3.25E-09

0.936 -1.05052 -1.22788 0.0049 -2.84E-09

0.944 -1.06034 -1.22788 0.0049 -2.47E-09

0.952 -1.07017 -1.22788 0.0049 -2.16E-09

0.96 -1.07999 -1.22788 0.0049 -1.88E-09

0.968 -1.08981 -1.22788 0.0049 -1.64E-09

0.976 -1.09964 -1.22788 0.0049 -1.43E-09

0.984 -1.10946 -1.22788 0.0049 -1.25E-09

0.992 -1.11928 -1.22788 0.0049 -1.09E-09

1 -1.1291 -1.22788 0.0049 -9.51E-10

1.008 -1.13893 -1.22788 0.0049 -8.29E-10

1.016 -1.14875 -1.22788 0.0049 -7.23E-10

1.024 -1.15857 -1.22788 0.0049 -6.31E-10

1.032 -1.1684 -1.22788 0.0049 -5.50E-10

1.04 -1.17822 -1.22788 0.0049 -4.80E-10

1.048 -1.18804 -1.22788 0.0049 -4.19E-10

1.056 -1.19787 -1.22788 0.0049 -3.65E-10

1.064 -1.20769 -1.22788 0.0049 -3.19E-10

1.072 -1.21751 -1.22788 0.0049 -2.78E-10

1.08 -1.22733 -1.22788 0.0049 -2.42E-10

1.088 -1.23716 -1.22788 0.0049 -2.12E-10

1.096 -1.24698 -1.22788 0.0049 -1.85E-10

1.104 -1.2568 -1.22788 0.0049 -1.61E-10

1.112 -1.26663 -1.22788 0.0049 -1.40E-10

1.12 -1.27645 -1.22788 0.0049 -1.22E-10

1.128 -1.28627 -1.22788 0.0049 -1.07E-10

1.136 -1.2961 -1.22788 0.0049 -9.32E-11

1.144 -1.30592 -1.22788 0.0049 -8.13E-11

1.152 -1.31574 -1.22788 0.0049 -7.09E-11



28

1.16 -1.32557 -1.22788 0.0049 -6.18E-11

1.168 -1.33539 -1.22788 0.0049 -5.40E-11

1.176 -1.34521 -1.22788 0.0049 -4.71E-11

1.184 -1.35503 -1.22788 0.0049 -4.11E-11

1.192 -1.36486 -1.22788 0.0049 -3.58E-11

1.2 -1.37468 -1.22788 0.0049 -3.12E-11

1.208 -1.3845 -1.22788 0.0049 -2.72E-11

1.216 -1.39433 -1.22788 0.0049 -2.38E-11

1.224 -1.40415 -1.22788 0.0049 -2.07E-11

1.232 -1.41397 -1.22788 0.0049 -1.81E-11

1.24 -1.4238 -1.22788 0.0049 -1.58E-11

1.248 -1.43362 -1.22788 0.0049 -1.38E-11

1.256 -1.44344 -1.22788 0.0049 -1.20E-11

1.264 -1.45327 -1.22788 0.0049 -1.05E-11

1.272 -1.46309 -1.22788 0.0049 -9.13E-12

1.28 -1.47291 -1.22788 0.0049 -7.97E-12

1.288 -1.48273 -1.22788 0.0049 -6.95E-12

1.296 -1.49256 -1.22788 0.0049 -6.06E-12

1.304 -1.50238 -1.22788 0.0049 -5.29E-12

1.312 -1.5122 -1.22788 0.0049 -4.61E-12

1.32 -1.52203 -1.22788 0.0049 -4.02E-12

1.328 -1.53185 -1.22788 0.0049 -3.51E-12

1.336 -1.54167 -1.22788 0.0049 -3.06E-12

1.344 -1.5515 -1.22788 0.0049 -2.67E-12

1.352 -1.56132 -1.22788 0.0049 -2.33E-12

1.36 -1.57114 -1.22788 0.0049 -2.03E-12

1.368 -1.58096 -1.22788 0.0049 -1.77E-12

1.376 -1.59079 -1.22788 0.0049 -1.55E-12

1.384 -1.60061 -1.22788 0.0049 -1.35E-12

1.392 -1.61043 -1.22788 0.0049 -1.18E-12

1.4 -1.62026 -1.22788 0.0049 -1.03E-12

1.408 -1.63008 -1.22788 0.0049 -8.95E-13

1.416 -1.6399 -1.22788 0.0049 -7.81E-13

1.424 -1.64973 -1.22788 0.0049 -6.81E-13

1.432 -1.65955 -1.22788 0.0049 -5.94E-13

1.44 -1.66937 -1.22788 0.0049 -5.18E-13

1.448 -1.6792 -1.22788 0.0049 -4.52E-13

1.456 -1.68902 -1.22788 0.0049 -3.94E-13

1.464 -1.69884 -1.22788 0.0049 -3.44E-13

1.472 -1.70866 -1.22788 0.0049 -3.00E-13

1.48 -1.71849 -1.22788 0.0049 -2.62E-13

1.488 -1.72831 -1.22788 0.0049 -2.28E-13

1.496 -1.73813 -1.22788 0.0049 -1.99E-13

1.504 -1.74796 -1.22788 0.0049 -1.74E-13

1.512 -1.75778 -1.22788 0.0049 -1.52E-13



29

1.52 -1.7676 -1.22788 0.0049 -1.32E-13

1.528 -1.77743 -1.22788 0.0049 -1.15E-13

1.536 -1.78725 -1.22788 0.0049 -1.01E-13

1.544 -1.79707 -1.22788 0.0049 -8.78E-14

1.552 -1.80689 -1.22788 0.0049 -7.66E-14

1.56 -1.81672 -1.22788 0.0049 -6.68E-14

1.568 -1.82654 -1.22788 0.0049 -5.83E-14

1.576 -1.83636 -1.22788 0.0049 -5.08E-14

1.584 -1.84619 -1.22788 0.0049 -4.43E-14

Modelling the motion of a falling styrofoam cup

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