12 x1 t07 01 projectile motion (2012)

Projectile Motion

Projectile Motiony

x

Projectile Motiony

x

45rangemaximum

Initial conditions

Projectile Motiony

x

45rangemaximum

Initial conditions when t = 0

v

Projectile Motiony

x

45rangemaximum


v

x

y

Projectile Motiony

x

45rangemaximum


v

x

y

cos

cos

vxvx

Projectile Motiony

x

45rangemaximum


v

x

y

cos

cos

vxvx

sin

sin

vyvy

Projectile Motiony

x

45rangemaximum


v

x

y

cos

cos

vxvx

sin

sin

vyvy

0x

Projectile Motiony

x

45rangemaximum


v

x

y

cos

cos

vxvx

sin

sin

vyvy

0x 0y

0x gy

1cx

0x gy

1cx 2cgty

0x gy

1cx 2cgty sin cos0,when vyvxt

0x gy


coscos1

vxvc

0x gy


coscos1

vxvc

sin

sin2

vgtyvc

0x gy


coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0x gy


coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

0x gy


coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

0x gy


coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

sin21

0

2

4

vtgty

c

0x gy 1cx 2cgty

sin cos0,when vyvxt

coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

sin21

0

2

4

vtgty

c

Note: parametric coordinates of a parabola

0x gy 1cx 2cgty

sin cos0,when vyvxt

coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

sin21

0

2

4

vtgty

c


cosxt

v

0x gy 1cx 2cgty

sin cos0,when vyvxt

coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

sin21

0

2

4

vtgty

c


cosxt

v

2

2 2

sin2 cos cos

gx xyv

22

2 sec tan2gxy xv

0x gy 1cx 2cgty

sin cos0,when vyvxt

coscos1

vxvc

sin

sin2

vgtyvc

3cos cvtx 42 sin

21 cvtgty

0 00,when yxt

cos03

vtxc

sin21

0

2

4

vtgty

c


cosxt

v

2

2 2

sin2 cos cos

gx xyv

22

2 sec tan2gxy xv

2

22 tan 1 tan

2gxy xv

Common Questions

Common Questions

(1) When does the particle hit the ground?

Common Questions

(1) When does the particle hit the ground?0when ground thehits Particle y

Common Questions


(2) What is the range of the particle?

Common Questions


(2) What is the range of the particle? 0 when find yi

Common Questions



xii intosubstitute

Common Questions



xii intosubstituteroots of the quadratic

Common Questions



xii intosubstitute(3) What is the greatest height of the particle?

Common Questions




0 when find yi

Common Questions




0 when find yi yii intosubstitute

Common Questions





roots of the quadratic

vertex of the parabola

Common Questions





(4) What angle does the particle make with the ground?

Common Questions





(4) What angle does the particle make with the ground? 0 when find yi

Common Questions






yii intosubstitute

Common Questions






yii intosubstitute

xyiii tan

x

y

Common Questions






yii intosubstitute

xyiii tan

x

y

roots of the quadratic

vertex of the parabola

(i) find slope of the tangent tanii m

Summary

SummaryA particle undergoing projectile motion obeys


0x gy and


0x gy and

with initial conditions


0x gy and


cosvx sinvy and


0x gy and


cosvx sinvy and

e.g. A ball is thrown with an initial velocity of 25 m/s at an angle ofto the ground. Determine;.

43tan 1


0x gy and


cosvx sinvy and


43tan 1

a) greatest height obtained


0x gy and


cosvx sinvy and


43tan 1

Initial conditionsa) greatest height obtained


0x gy and


cosvx sinvy and


43tan 1


43tan 1

3

4

5


0x gy and


cosvx sinvy and


43tan 1


43tan 1

3

4

5cosvx


0x gy and


cosvx sinvy and


43tan 1


43tan 1

3

4

5

m/s205425

x

cosvx


0x gy and


cosvx sinvy and


43tan 1


43tan 1

3

4

5

m/s205425

x

sinvy cosvx


0x gy and


cosvx sinvy and


43tan 1


43tan 1

3

4

5

m/s205425

x

sinvy cosvx

m/s155325

y

0x 10y

0x 10y

1cx

0x 10y

1cx 210 cty

0x 10y

1cx 210 cty 15 200,when yxt

0x 10y


20201

xc

0x 10y


20201

xc 1510

152

tyc

0x 10y


20201

xc 1510

152

tyc

320 ctx

0x 10y


20201

xc 1510

152

tyc

320 ctx 42 155 ctty

0x 10y


20201

xc 1510

152

tyc

320 ctx 42 155 ctty 0 00,when yxt

0x 10y


20201

xc 1510

152

tyc


txc

2003

0x 10y


20201

xc 1510

152

tyc


txc

2003

tty

c

155

02

4

0x 10y


20201

xc 1510

152

tyc


txc

2003

tty

c

155

02

4

0when occursheight greatest y

0x 10y


20201

xc 1510

152

tyc


txc

2003

tty

c

155

02

4


23

01510

t

t

0x 10y


20201

xc 1510

152

tyc


txc

2003

tty

c

155

02

4


23

01510

t

t

445

2315

235,

23when

2

yt

0x 10y


20201

xc 1510

152

tyc


txc

2003

tty

c

155

02

4


23

01510

t

t

445

2315

235,

23when

2

yt

ground theabove m4111 isheight greatest

b) range

b) range

time of flight is 3 seconds

b) range


06 320,3when

xt

b) range


06 320,3when

xt

60m is range

b) range


06 320,3when

xt

60m is range

second21after ball theofdirection and velocity c)

b) range


06 320,3when

xt

60m is range


01

152110 20,

21when

yxt

b) range


06 320,3when

xt

60m is range


01

152110 20,

21when

yxt

20

10510

b) range


06 320,3when

xt

60m is range


01

152110 20,

21when

yxt

20

10510

432621tan

b) range


06 320,3when

xt

60m is range


01

152110 20,

21when

yxt

20

10510

432621tan

horizontal the to4326 of anglean

at travelingisit and m/s510 velocity second,21after

d) cartesian equation of the path


20

20xt

tx


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty

Using the cartesian equation to solve the problem


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty


a) greatest height is y value of the vertex


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty


a) greatest height is y value of the vertex23 14( )(0)

4 809

16


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty



4 809

16

49 20

16 1454

ya


20

20xt

tx

43

80

2015

205

155

2

2

2

xxy

xxy

tty



4 809

16

49 20

16 1454

ya

ground theabove m4111 isheight greatest

b) range

b) range2 3

80 4

34 20

x xy

x x

b) range2 3

80 4

34 20

x xy

x x

roots are 0 and 6060m isrange

b) range2 3

80 4

34 20

x xy

x x



b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

cosxv

t

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

cosxv

t

2

15

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

cosxv

t

2

15

101 22 5

10 5

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

cosxv

t

2

15

101 22 5

10 5

1after second, velocity 10 5m/s and it is2

traveling at an angle of 26 34 to the horizontal

Exercise 3G; 1ac, 2ac, 4, 6, 8, 9, 11, 13, 16, 18

Exercise 3H; 2, 4, 6, 7, 10, 11

b) range2 3

80 4

34 20

x xy

x x



1when , 102

t x 2 380 4

340 4

x xy

dy xdx

1when 10,2

dyxdx

432621tan

cosxv

t

2

15

101 22 5

10 5

1after second, velocity 10 5m/s and it is2

traveling at an angle of 26 34 to the horizontal

12 x1 t07 01 projectile motion (2012)

Education

Transcript of 12 x1 t07 01 projectile motion (2012)