Add Maths SBA

7/24/2019 Add Maths SBA

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Additional Mathematics SBA


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ContentsProject Title.................................................................................................................2

Problem...................................................................................................................... 2

Assumptions............................................................................................................... 2

Description of Problem Areas.................................................................................. 2

Hole 1!................................................................................................................2

Hole 1"!................................................................................................................2

Hole #!..................................................................................................................2

Mathematical formulation...........................................................................................2

Problem Solution........................................................................................................ 2

Calculations to determine a ball beha$ior about launch.........................................2

Calculations To determine ball beha$ior on %mpact and on the &reen.................2

Application of Solution................................................................................................ 2

Hole 1.................................................................................................................... 2

Stro'e 1 !.............................................................................................................. 2

Stro'e 2!............................................................................................................... 2

Stro'e ( !.............................................................................................................. 2

Hole 1".................................................................................................................... 2

Stro'e 1!............................................................................................................... 2

Stro'e 2................................................................................................................ 2

Stro'e (................................................................................................................ 2

Hole #...................................................................................................................... 2

Stro'e................................................................................................................... 2

Discussion of )indin&s................................................................................................ 2

Conclusion.................................................................................................................. 2

Project TitleTheoretical calculations in the 'inematics of &ol*n&.


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ProblemA &olfer +ants to &et determine +hich &olf club and +hat speed he must impart on a &olf

ball in order to &olf club ,s+in& speed- to project the ball *rstl/ from the teein& area

onto the fair+a/ then from the fair+a/ onto the &reen and *nall/ from the &reen to the

hole +ithin a ma0imum of " sto'es per hole ha$in& its o+n limits in the number of stro'es

do+n the dierent holes. Purpose of stud/

The purpose of the stud/ is to determine the an&le to the horiontal at +hich the ball must the ballmust be struc' the $elocit/ at +hich the &olfer must s+in& his club and the t/pe of club to be usedusin& 'inematics to accomplish clearin& the hole. 3ssential considerations include

4e&lectin& air resistance and +ind direction and usin& the $alue of &ra$it/ as 15ms61 .

The trajector/ of the &olf ball

The launch an&le must be measured to a relati$e horiontal plane

The ball is struc' at &round le$el and the &olf course has a relati$el/ lo+ relief

The &olfer is allo+ed a sin&le reposition per hole of no more than 15 meters behind

the end of his stro'e. The reposition counts as a stro'e.

7olf club 8aunch An&le Ma0 Distance9 %ron :" 125m; %ron (5< 1:;m" %ron 2"< 1;"mDri$er 11< 22;mPitchin& =ed&e :< 25611" /ds.

AssumptionsAs research has sho+n the ener&/ at +hich the ball stri'es the &round is partiall/ absorbed b/ the&rass ,the *eld +ill absorb 9"> of its jinect so that the bounce recoil is 2"> of the hei&ht of

it hittin& the &round-

The ball is rebounded +ith a 2"> of its to 'inetic ener&/ as +ell.

The coe?cient of friction of the &rass is said to be at .(

The &olfer is capable of s+in& his &olf club at 5;5:5and (5 m

The &olf clubs loft +ill be used as the launch an&le and the &olf clubs he uses e0cludin& his putterare sho+n belo+

The ener&/ bet+een the club and the ball is completel/ transferred

Description of Problem Areas


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Hole 1!This hole is "25m across at the furthest tee bo0 to the.

Calculations in this hole +ill be dictated b/ &i$en $alues fordistance and &i$en launch an&les!

)actors to consider

6 This +ill be done in " sto'es6 @nl/ @ne stro'e +ill be done on the &reen6 The ball +ill not cross the clis6 The &reen be&ins at the "15m mar'


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Hole 1"!

This hole (;2m across at the furthest teebo0 to the hole.

This hole contains ( notable haards a minor sand traplocated on the path+a/ in +hich the &olfer plans to project hisball a fe+ trees on the path +a/ as +ell as folia&ebloc'in& the tee bo0. Calculations in this hole +ill resol$e+ith &i$en $elocities and launch an&le.

6 )or this hole the reuired number of stro'es

+ill be determined b/ the trajectories used

b/ the &olfer.

Hole #!This Hole +ill be done in 1 stro'e for a hole in one. This is

the smallest hole in the set onl/ measurin& 9; meters +ith

1 main haards. This bein& a rather lar&e sand trap in the

path+a/ of the ball. As onl/ one stro'e +ill be used the onl/

&olf club that +ill be used on this hole +ill be the Dri$er.


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Mathematicalformulation


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1- To sho+ that if the &olf ball is projected at a certain an&le at a $elocit/ then the

horiontal component is &i$en b/ y=Vosin and the $ertical component is &i$en

b/ b/ x=Vocos

2- To use 4e+tons 3uations of motion

2

2

1

222

atuts

asuv

atuv

+=

+=

+=

(- To use the la+s of conser$ation of momentum and ener&/ the ball $elocit/ can be

calculated as &i$en b/ the follo+in& formulas

k=1

2 mv

2

M1V1=M2V2

Horiontal Component

ertical Component


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:- The Eelationship bet+een the $ertical component and horiontal component is

sFutG12 atI2. Jsin& the formulas for the $ertical component and horiontal

component the euation can be no+ made into y=xtan1

2g( xvcos)

2

This euation is in the form a0I2 G b0 G c and therefore +hen plotted +ill produce

a cur$e as sho+n belo+


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ProblemSolution


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Calculations to determine a ball

beha$ior about launch%n the precedin& points the formulation is &i$en to determine the trajector/ of the ball to

tra$el from the tee o to the areas on the fair+a/ and to tra$erse the fair+a/ alon& +ithcrossin& haards on the multitude of *elds.

%f the &olf ball is projected at a theta an&le and at a $elocit/ of then the horiontal component or

speed at +hich it mo$es horiontall/ is &i$en b/

sinoVy =

and the $ertical component or the

$elocit/ at +hich the ball mo$es at $erticall/ is &i$en b/ x=Vocos . As sho+n the b/ the

belo+ &raph

As sho+ in the follo+in& calculations the euations can be resol$ed

cos= Adjacent

hypotenuse

=here the $elocit/ is the h/potenuse and the $ertical component is the adjacent $ector.

The formula no+ becomes

cos=vertical component(x )/ velocity (v)

The formula can no+ be rearran&ed to *nd the $ertical component of $elocit/

vcos=Vx

The horiontal component can also be resol$ed usin& the similar method in +hich


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sin= opposite

hypotenuse

=here opposite is the horiontal component or distance at +hich it mo$es horiontall/

and h/potenuse is the $elocit/. The euation can be rearran&ed in a similar +a/ to

produce

vsin=Vy

Jsin& the 'inematic euations the $arious $elocities and distances needed can be

calculated

221

222

atuts

asuv

atuv

+=

+=

+=

The Eelationship bet+een the $ertical component and horiontal component is

s=ut+1

2a t

2

8ettin& s be the horiontal distance and substitutin& the $elocit/ of the / or

horiontal component as the initial $elocit/ and the $elocit/ for &ra$it/ as the *nal

$elocit/. The euation no+ becomes

y=(vsin ) t(1 /2)>2

As &ra$it/ is not applied for the 0 component the distance at +hich the ball tra$els +ith respect to$ertical component is &i$en b/ x=vcos t

tis made the subject of the formula t= x

vcos

Substitutin& the deri$ed euation for t into the pre$ious formula


13/14

y=( xvcos ) vsin1

2g( xvcos)

2

The euation could be further simpli*ed into

y=xtan 1

2g ( xvcos)

2

The euation is in the form of y=ax2+bx+c and therefore +hen plotted on the Cartesian plane

produces as cur$e sho+n belo+. This sho+s that the ball tra$els alon& a parabolic path. As sho+nbelo+. As sho+n b/ the dia&ram +here the ball lands is the position at +hich the ball hits the&round.


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L m

2"> Lm F L2 m

2"> L2 m

Calculations To determine ball beha$ior

on %mpact and on the &reen)or the simpli*cation of calculations the ball assumed to ha$e rebounded +ith appro0imatel/ 2">of its ma0 fallin& distance and appro0imatel/ 2"> of its 'inetic ener&/ meanin& that on impact theball +ill rebound into the air at a fraction of of its speed and po+er as demonstrated b/ thedia&ram =hen the rebound hei&ht of the ball is /r5 cm the ball +ill be allo+ed to enter itsrollin& state. The fall distance is the / coordinate of the stationar/ point.

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Add Maths SBA