1.0: Introduction: Recently I got an opportunityto witness an event in which professional skydivers were performing dangerousstunts at extremely high altitudes.
I was awestruck after watching them performso precisely without being scared of heights, or maybe they were. I was amazedby watching them and that event was in my mind for a lot of days. I can stillrecollect fondly the questions which popped into my head at that time; Will thespeed of their fall increase continuously? Does the size/shape/mass of theperson affect their path? What could be the minimum height from the ground atwhich they could deploy their parachutes? Etc. Whilethinking about a good topic for my exploration I thought of using this event asa premise for my investigation.
I am more of a theorist when it comes tomathematics rather than the application part involved in it, since theconfusion faced in solving problems makes it more interesting for me whencompared to applying them in real life. However, this case was not really abouttheory or application, instead it was about exploring something interesting. On further research, I found that I canuse calculus and few concepts in physics to model equations which describe themotion of any freely falling object. My aim is to model freely falling objectsof different size, large and small objects to be specific.
To calculate whatcan be called small and large object in modelling freely falling objects mightbecome a type of investigation, which will not be explored in a great depth atall in this research. Freely falling objects are very commonto everyone and they are visible very frequently to the naked eye. In fact, oneof the most famous example is of an apple falling on Newton’s head, whichinspired him to discover the gravity and henceforth the gravitational constant.Gravity and Newton’s second law of motion are important concepts involved anddealt with in this research.
I have classified the scenario intothree 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 tohave a prior knowledge of newton’s second law: The acceleration ofan object as produced by a net force is directly proportional to the magnitudeof the net force, in the same direction as the net force, and inversely proportionalto the mass of the object1. Let usconsider objects with different size and masses. The variables are as follows: 1.
1: Objects falling in vacuum For anyobject falling in the vacuum, irrespective of the mass or size, a free bodydiagram 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: Theabove equation of velocity is similar to the linear equation which is in theform of:Sincethe objects falling in vacuum are in the form of a linear equation, their pathis like a straight line shown below. Figure C 1.2 Small Objects falling against air resistance. For anysmall object falling against air resistance their free body diagram would looksomething just like this: mgFigure D When asmall object falls downwards, the forces which determine its velocity andacceleration depend on many factors out of which ‘mass’ and ‘gravity’ are theimportant ones. These will increase its acceleration and drag force (resistiveforce) will decrease the acceleration. As we see that ‘mg’ is a vector goingdownwards so it’s positive (+) whereas the drag force is acting against theobject (upwards) hence it is negative (-). Drag force:Forsmaller objects, drag force is directly proportional to velocity3 Ø ‘b’ is consideredas the proportionality constant. Using Newton’s second law, we get: By re arrangingthe equation, we get: Integrating onboth sides we get: Therefore, bysubstituting: Hence, we get: Making ‘V’ as thesubject we derive to the following equation: Thistype of graph suggests that the path of the object might be exponential, suchas in the figure below: Figure E 1.
3 Large Objects falling against air resistance For anysmall object falling against air resistance their free body diagram would looksomething just like this: mg Figure F When a largeobject falls downwards then the forces which determine its velocity andacceleration depend on many factors out of which ‘mass’ and ‘gravity’ willincrease its acceleration and drag force (resistive force) will decrease theacceleration. 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 isnegative (-). Drag force for larger objects, accordingto secondary sources, was found to be directly proportional to , hence Ø ‘d’ isused as the proportionality constant here:UsingNewton’s second law, we get By re-arrangingwe get: Integrating bothsides we get: At we get Makin’v’ as the subject we get: Eventhis equation appears to be an exponential graph somewhat like this. Figure GEXPERIMENT: Keeping in mind the above inferences, Iwas intrigues by the results of how each object’s model would look like. So, Idecided to look for it myself.
I thought of conducting an experiment in whichI’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 withsponge 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, myhypothesis is that since sponge ball fits in the category of small object, andthrowing it from the terrace of my building would encourage drag force toinfluence its path, the velocity-time graph of the fall would be exponential. Let’s move on to the main findings andresults:Figure H is a screenshot of a part ofvideo where the ball is shown through an arrow mark. Figure HThe next figure shows how a trackingsoftware 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 wegraph these values, the model of the path of sponge ball can be determined.
Thefollowing 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 bodydiagram(FBD): FBD is a visual illustration of all the different forces actingon an object at any given instant.3 “Newton’s Second Law.” Physicsclassroom.Com,2018,http://www.