Modelling shuttlecock and looking into the equation

Modelling the flight trajectory of a
shuttlecock

Aim

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Introduction

Badminton is a unique
sport which uses a unique shaped object, the shuttlecock. Unlike other racket
sports which uses a regularly shaped ball as the object, the shuttlecock has a
special structure. Shuttlecocks are made with different types of materials such
as plastic and feather. Although in professional play, almost only the feather
type is used. It has the shape of a frustum of a cone that is mounted that is
held together by a round center, usually made of cork or rubber. The cone-shape
shape the feathers.

The unique shape of
the shuttlecock allows it to have a steep flight trajectory and generates a lot
of aerodynamic force. Because of this, the trajectory that it makes is unsymmetrical
and differs from the typical mathematical parabola.

The sport itself is
very popular in the country that I live in, Indonesia. Not only is it one of
the most popular sport in the country, it is also Indonesia’s most successful
sport. Countless players from Indonesia have won world championships and
Olympic medals at the sport. The sport is played by almost everyone from all
age groups professionally and leisurely. By studying and analyzing the flight
of a shuttlecock and looking into the equation of the it, training for players
could be done more effectively on how they can hit a shuttlecock that may be
harder for the opponent to return.

Recently, in my
mathematics class, I have learnt about the powers of calculus. With the power
of calculus, lots of things can be done such as finding the equation of acceleration
and the displacement, by either integrating and differentiating an equation.
With this, I have learnt that an equation for the flight of the shuttlecock can
be derived.

Deriving the equation

The terminal velocityra2 

 

To derive the
equation, we first have to learn about the physics behind a shuttlecock. When a
shuttlecock is in flight, the unique shape allows the air resistance or drag
force, a force that opposes the direction of the shuttlecock to play a huge
part in affecting the flight of it. Any object follows newton’s second law of
motion which is,

 

Where F is the force,
m is the mass and a the acceleration. The force has lots of components, there
is buoyancy, denoted by B, the weight of the object, denoted by mg, and also
the drag force Fd. This force can also be written as

and  are values of
the coefficient of the shuttlecock that can be determined experimentally and is the terminal velocity. So the first equation can
also be written as:

The buoyancy or B in the air and shuttlecock is very
small compared to the weight and drag force so it can be neglected and taken
out of the equation. The terminal velocity is when the weight and the
resistance force of an object balances and cancels each other out. Like  in the diagram below

 

 

 

 

 

                                    

Next we have
to make an expression for the vertical and horizontal component when the
shuttlecock is moving. If  was the initial
velocity of the shuttlecock when it was hit and the angle of elevation of the
shuttlecock is , the horizontal and vertical component off the
velocity would be, and  because it
follows the rules of trigonometry like shown belowra4 .

 

           

 

 

 

Going back to the previous equation with n. typically at low velocities a
value of n=1 is used and at high velocities, n=2 is used. since in the sport of
badminton, it is hit at both high speed and low speed, there are 2 equations
that can be derived, one with an n value of 1, and another with a value of 2.

      I.        
At n=1

 

 

  

 ra3Define all formulae

1.     variables being used

2.     equations and words must properly deifined.

 ra4No personal engagement seen yet

 

What reflection can be drawn from your
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