1.0: research. Freely falling objects are very common to

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.

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 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.