1.0: Introduction:

Recently I got an opportunity

to witness an event in which professional skydivers were performing dangerous

stunts at extremely high altitudes. I was awestruck after watching them perform

so precisely without being scared of heights, or maybe they were. I was amazed

by watching them and that event was in my mind for a lot of days. I can still

recollect fondly the questions which popped into my head at that time; Will the

speed of their fall increase continuously? Does the size/shape/mass of the

person affect their path? What could be the minimum height from the ground at

which they could deploy their parachutes? Etc.

While

thinking about a good topic for my exploration I thought of using this event as

a premise for my investigation. I am more of a theorist when it comes to

mathematics rather than the application part involved in it, since the

confusion faced in solving problems makes it more interesting for me when

compared to applying them in real life. However, this case was not really about

theory or application, instead it was about exploring something interesting.

On further research, I found that I can

use calculus and few concepts in physics to model equations which describe the

motion of any freely falling object. My aim is to model freely falling objects

of different size, large and small objects to be specific. To calculate what

can be called small and large object in modelling freely falling objects might

become a type of investigation, which will not be explored in a great depth at

all in this research.

Freely falling objects are very common

to everyone and they are visible very frequently to the naked eye. In fact, one

of the most famous example is of an apple falling on Newton’s head, which

inspired him to discover the gravity and henceforth the gravitational constant.

Gravity and Newton’s second law of motion are important concepts involved and

dealt with in this research.

I have classified the scenario into

three different cases:

· Objects

falling in vacuum.

· Small

Objects falling against air resistance.

· Large

Objects falling against air resistance.

Figure

A

In order to proceed further we need to

have a prior knowledge of newton’s second law:

The acceleration of

an object as produced by a net force is directly proportional to the magnitude

of the net force, in the same direction as the net force, and inversely proportional

to the mass of the object1.

Let us

consider objects with different size and masses. The variables are as follows:

1.1: Objects falling in vacuum

For any

object falling in the vacuum, irrespective of the mass or size, a free body

diagram at an instant is given in figure 1:

mg

Figure B2

Using newton’s second law, we get:

(Cancelling ‘m’ on both sides)

Integrating on both sides, we get:

The

above equation of velocity is similar to the linear equation which is in the

form of:

Since

the objects falling in vacuum are in the form of a linear equation, their path

is like a straight line shown below.

Figure C

1.2 Small Objects falling against air resistance.

For any

small object falling against air resistance their free body diagram would look

something just like this:

mg

Figure D

When a

small object falls downwards, the forces which determine its velocity and

acceleration depend on many factors out of which ‘mass’ and ‘gravity’ are the

important ones. These will increase its acceleration and drag force (resistive

force) will decrease the acceleration. As we see that ‘mg’ is a vector going

downwards so it’s positive (+) whereas the drag force is acting against the

object (upwards) hence it is negative (-).

Drag force:

For

smaller objects, drag force is directly proportional to velocity3

Ø

‘b’ is considered

as the proportionality constant.

Using Newton’s second law, we get:

By re arranging

the equation, we get:

Integrating on

both sides we get:

Therefore, by

substituting:

Hence, we get:

Making ‘V’ as the

subject we derive to the following equation:

This

type of graph suggests that the path of the object might be exponential, such

as in the figure below:

Figure E

1.3 Large Objects falling against air resistance

For any

small object falling against air resistance their free body diagram would look

something just like this:

mg

Figure F

When a large

object falls downwards then the forces which determine its velocity and

acceleration depend on many factors out of which ‘mass’ and ‘gravity’ will

increase its acceleration and drag force (resistive force) will decrease the

acceleration. As we see that ‘mg’ is a vector going downwards so it’s positive

(+) whereas the drag force is acting against the object (upwards) hence it is

negative (-).

Drag force for larger objects, according

to secondary sources, was found to be directly proportional to , hence

Ø

‘d’ is

used as the proportionality constant here:

Using

Newton’s second law, we get

By re-arranging

we get:

Integrating both

sides we get:

At we get

Makin

‘v’ as the subject we get:

Even

this equation appears to be an exponential graph somewhat like this.

Figure G

EXPERIMENT:

Keeping in mind the above inferences, I

was intrigues by the results of how each object’s model would look like. So, I

decided to look for it myself. I thought of conducting an experiment in which

I’d throw a ball from the terrace of my house and model the path of its fall.

For conducting the experiment, I chose a sponge ball; playing cricket with

sponge ball was quite difficult because no matter how fast you try and throw,

its speed would always be lesser than compared to cork ball. Therefore, my

hypothesis is that since sponge ball fits in the category of small object, and

throwing it from the terrace of my building would encourage drag force to

influence its path, the velocity-time graph of the fall would be exponential.

Let’s move on to the main findings and

results:

Figure H is a screenshot of a part of

video where the ball is shown through an arrow mark.

Figure H

The next figure shows how a tracking

software graphs the path of the ball and generates the values of x and y axes.

Figure I

Values:

Mass_A

t

V(y)

0.00E+00

3.38E-02

-1.12E+03

6.76E-02

-1.23E+03

1.01E-01

-1.29E+03

1.35E-01

-1.42E+03

1.69E-01

-1.48E+03

2.03E-01

-1.47E+03

2.36E-01

-1.55E+03

2.70E-01

-1.61E+03

3.04E-01

-1.63E+03

3.38E-01

-1.68E+03

3.72E-01

-1.75E+03

4.05E-01

-1.82E+03

4.39E-01

-1.85E+03

4.73E-01

-1.85E+03

5.07E-01

-1.88E+03

5.40E-01

-1.96E+03

5.74E-01

-1.99E+03

6.08E-01

-2.02E+03

6.42E-01

-2.07E+03

6.76E-01

-2.12E+03

7.09E-01

-2.32E+03

7.43E-01

-2.31E+03

7.77E-01

-2.17E+03

8.11E-01

-2.24E+03

8.44E-01

-2.28E+03

8.78E-01

-2.27E+03

9.12E-01

-2.31E+03

9.46E-01

-2.36E+03

When we

graph these values, the model of the path of sponge ball can be determined. The

following figure is of the graph of the above values.

1 “Newton’s Second Law.” Physicsclassroom.Com,

2018, http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law.

2 A free body

diagram(FBD): FBD is a visual illustration of all the different forces acting

on an object at any given instant.

3 “Newton’s Second Law.” Physicsclassroom.Com,

2018,

http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law.