mechanics lecture 2, slide 1 vectors and 2d-kinematics continued relevant equations how to use them...

Mechanics Lecture 2, Slide 1

Vectors and 2d-KinematicsContinued

Relevant Equations How to use them Homework Hints

Hyperphysics-Trajectories


http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html


Projectile Motion Quantities


Initial velocityspeed,angle

Maximum Height of trajectory, h=ymax

Range of trajectory, D

Height of trajectory at arbitrary x,t

“Hang Time”Time of Flight, tf

Maximum Height of Trajectory


Height of trajectory,h=ymax

g

vyh

g

vyh

g

v

g

vyh

gttvyh

tyh

g

vtvy

gtvtv

y

yy

yyy

y

yyy

yy

2

sin

2

2

1

2

1

)(

0

)(

220

0

20

0

20

20

0

200

0max

0

maxmax

max

max

Time of Flight


Time of Flight, “Hang Time”


g

v

g

vtt

gtvtgttv

gttvyy

yttytt

gttvyty

y

yy

y

y

ff

ffff

ff

ff

sin22;0

02

1

2

12

1

)(2

1)(

00

02

0

2000

0

200

Range of trajectory


g

vD

g

v

g

vvD

tvttxD

yttytt

tvxtx

yx

fxf

ff

x

2sin

sincos22

0)(

)(

)(

20

2000

0

0

00

Angle for Maximum Range


0

0

45

902

0)2cos(0)(

)2cos(2)(

)2sin()(

d

dfd

df

f

MAXIMUM range OCCURS AT 450

Will it clear the fence


Height of Trajectory at time t or position x


200 2

1)( gttvyty y

Height of trajectory, y(x)

x

x

v

xxt

tvxx

0

0

00

Height of trajectory, y(t)

2

000

2

000

00

2

0

0

0

000

cos2

1

cossin)(

2

1)(

0;0

2

1)(

v

xg

v

xvxy

v

xg

v

xvxy

yx

v

xxg

v

xxvyxy

xxy

xxy

Projectile Trajectory Equations


2

000 cos2

1

cossin)(

v

xg

v

xvxy

200 2

1)( gttvyty y

Height of trajectory as f(x), y(x)

Height of trajectory as f(t) , y(t)

g

vD

2sin20

g

v

g

vt y

f

sin2200

g

vyh

2

sin220

0

Range of trajectory

Time of Flight (“Hang Time”)

Maximum height

Where will it land?


Launch Velocity-Given R and


Launch Angle


Launch Velocity –Given R and h



Field Goal ExampleA field goal kicker can kick the ball 30 m/s at an angle of 30 degrees w.r.t. the ground. If the crossbar of the goal post is 3m off the ground, from how far away can he kick a field goal?

y-direction

voy = vo sin(30o) = 15 m/s

y = yo + voyt + ½ at 2

3 m = 0 m + (15 m/s) t – ½ (9.8 m/s2) t 2

t = 2.8 s or t = 0.22 s.

x-direction

vox = vo cos(30o) = 26 m/s

D = xo + vox t + ½ at 2

= 0 m + (26 m/s)(2.8 s) + 0 m/s2 (2.8 s )2

= 72.8 m

D

3 m

y

x

Illini Kicks 70 yard Field Goal

http://www.youtube.com/watch?v=VvT9wvWHFM8

Homework Hints-Baseball


Homework Hints- Baseball Stadium Wall


Homework Hints – Stadium Wall


Calculate time to reach wall using vx:

cos// 00 vxvxt wallwallwall x

Calculate y position at time to reach wall:

200

20000

200

cos/2

1tan

cos/2

1cos/sin

2

1

vxgxyy

vxgvxvyy

tgtvyy

wallwallwall

wallwallwall

wallwallwall y

Homework Hints-Catch




cos00 vvx

sin00 vvx

g

vy

g

vyy y

2

sin

2

)( 20

0

20

0max

g

v

g

vvtvx

g

vt

tvtgtgtv

tgtvyyyy

yx

x

y

yy

y

ff

f

ffff

ffff

sincos22

22

1;

2

10

2

1;

200

0

0

022

0

2000

0



20000

0000

0

max1

0

max

/2

1/

cos;sin

)(cos;

)(cos

xjuliejuliejulie vxgvxvyy

vvvv

v

yyv

v

yyv

xy

xy

max2

0max0

20

20

20

max00max2

0

2

cossin

cos;2sin

yyvyygv

vvv

yyvvyygv

Homework Hints-Catch 2




cos0vvx vVx is constant !

g

vgvtgvttv y

yy ffy

0

00

2)(

g

vt

tvtgtgtv

y

yy

f

ffff

0

022

0

22

1;

2

10

Kinetic energy should be same as when ball was thrown. Y-component of velocity would be downward.



g

vvtvx yx

x ff

00

0

2

julie

julie

t

xv

x0

Same conditions as before

max2

0max0 2 yyvyygv

20000 /2

1/ xjuliejuliejulie vxgvxvyy

xy

Homework Hints – Soccer Kick & Cannonball




20

200 yx

vvv

x

y

v

v

0

01tan

g

vyy y

2

)( 20

0max

g

vD

2sin2

0

g

vvtvx yx

x ff

00

0

2



)()(

)(;)(

220

00

givenyxgiven

givenxgivengiveny

ttvvttv

vttvtgvttvxy

200 2

1givengivengiven tgtvytty

y

Trigonometric Identity for range equation


)2sin(2

1)sin()sin(

2

1cossin

)sin()sin(2

1cossin

222

1cossin

4cossin

422cossin

2cos

2sin

)()()()(

)()()()(

i

ee

i

ee

i

eeee

i

eeeeeeeeee

i

ee

ee

i

ee

iiii

iiii

iiiiiiiiiiii

ii

ii

http://mathworld.wolfram.com/Cosine.html http://mathworld.wolfram.com/Sine.html

http://mathworld.wolfram.com/Cosine.html

http://mathworld.wolfram.com/Sine.html

Trigonometric Identities relating sum and products


List of trigonometric identities

cossin2sincoscossin)2sin(

sincoscossin)sin(

http://www.ask.com/wiki/List_of_trigonometric_identities?o=2801&qsrc=999&ad=doubleDown&an=apn&ap=ask.com#Angle_sum_and_difference_identities

Question 2


mechanics lecture 2, slide 1 vectors and 2d-kinematics continued relevant equations how to use them...

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