Modelling the flight trajectory of a

shuttlecock

Aim

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

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