Projectilemotion

Kinematics in 2d

dv

dt= a

d!v

dt=!a

dvx

dt= a

x

dvy

dt= ay

one dimension:

two dimensions:

Kinematics in 2d

onedimension:

twodimensions:

x ! xi= v

it +

1

2at

2

!r !!ri=!vit +

1

2

!a t

2x ! x

i= v

xit +

1

2axt2

y ! yi = vyit +1

2ay t

2

Decomposing motion

vyi

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

a = -gj

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* vertically upward

*given initial upward velocity viy

1d motion of ball:

Decomposing motion

vx=Vo

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

Car pushed* rightward

*given initial rightward velocity Vo

1d motion of car:

Decomposing motion

vx=Vo

vyi vi

vxi=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

Ball launched* from moving car

*given initial velocity vi=Voi + viyj

2d motion of ball:

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

Decomposing motion

vx=Vo

a = -gj

�

x(t) = xi

+ vxit +

1

2axt2

= xi

+Vot

�

y(t) = yi + vyit +1

2ayt

2= vyit !

g

2t2

2d motion of ball:same as car

classic range problemviy

xθ An outfielder throws a ball at

speed |vi|=30 m/s at an angleθ=30o. How far does it travel?

θ

vi

vxi=30 cos(30o) = 26 m/s

vyi=30 sin(30o) = 15 m/s

1. Decompose vi

2. Analyze one component

A. x-motion for xfB. x-motion for vxfC. y-motion for tD. y-motion for vyf

�

y f = yi + vyit !1

2gt

2

0 = 0 + 15 t - 4.9 t2

0 = t (15 - 4.9 t )

tf = 15/4.9 = 3.1 s

classic range problemviy

xθ An outfielder throws a ball at

speed |vi|=30 m/s at an angleθ=30o. How far does it travel?

θ

vi

vxi=30 cos(30o) = 26 m/s

vyi=30 sin(30o) = 15 m/s

1. Decompose vi

2. Analyze one component

A. x-motion for xfB. x-motion for vxfC. y-motion for tD. y-motion for vyf

tf = 15/4.9 = 3.1 s

3. Analyze other component

�

x f = xi + vxit +1

2axt

2

xf = 0 + 26 tf +0 = 81 m

classic range problemviy

xθ An outfielder throws a ball at

speed vi at an angle θ. Howfar does it travel?

θ

vi

vxi=vi cos(θ)

vyi=vi sin(θ)

1. Decompose vi

2. Analyze one component

A. x-motion for xfB. x-motion for vxfC. y-motion for tD. y-motion for vyf

�

y f = yi + vyit !1

2gt

2

�

0 = t[visin(!) "1

2gt]

�

t f =2visin(!)

g

classic range problemviy

xθ An outfielder throws a ball at

speed vi at an angle θ. Howfar does it travel?

θ

vi

vxi=vi cos(θ)

vyi=vi sin(θ)

1. Decompose vi

2. Analyze one component

A. x-motion for xfB. x-motion for vxfC. y-motion for tD. y-motion for vyf

�

y f = yi + vyit !1

2gt

2

�

0 = t[visin(!) "1

2gt]

�

t f =2visin(!)

g

3. Analyze other component

�

x f = xi + vxit +1

2axt

2

�

x f = vi cos(!)t f

�

x f =vi2

g2 sin(!) cos(!) =

vi2

gsin(2!)

Old Exam ProblemA rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

70 m/s

0

40 m

xfx

A. x-motion for tB. x-motion for vxfC. y-motion for tD. y-motion for vyf

First step is

Which Equation would you use?

ABCD�

vyf = vyi + ayt

�

y f = yi + vyit +1

2ayt

2

�

y f = yi +1

2(vyf + vyi)t

�

vyf2

= vyi2

+ 2ay (y f ! yi)

Old Exam Problem

0 xfx

�

0 = 40m!1

2gt f

2

�

t f =80m

9.8m/s2

= 2.9s

A rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

70 m/s

40 m

Which Equation would you use?

ABCD�

vxf = vxi + axt

�

x f = xi + vxit +1

2axt

2

�

x f = xi +1

2(vxf + vxi)t

�

vxf2

= vxi2

+ 2ax (x f ! xi)

Old Exam Problem

0 xfx

�

0 = 40m!1

2gt f

2

�

t f =80m

9.8m/s2

= 2.9s

�

x f = vxit f = 200m

A rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

70 m/s

40 m

A rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

Old Exam Problem

0 xfx

70 m/s

40 m

�

t f =80m

9.8m/s2

= 2.9s

Which Equation would you use?

ABCD�

vyf = vyi + ayt

�

y f = yi + vyit +1

2ayt

2

�

y f = yi +1

2(vyf + vyi)t

�

vyf2

= vyi2

+ 2ay (y f ! yi)

Old Exam Problem

0 xfx

�

vyf = !gt f = !28m/s

A. v=28 m/sB. v=70 m/sC. v=75 m/sD. v=98 m/s

�

v = (70m/s)2

+ (!28m/s)2

= 75m/s

A rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

70 m/s

40 m

�

t f =80m

9.8m/s2

= 2.9s

A rescue plane is flying level at 40 m above the water, at a speed of 70 m/s.The pilot wishes to drop a deflated life raft to a person in the water.a. At what distance from his target should he release the raft?b. At what speed will the package hit the water?c. At what angle will the package hit the water?

Old Exam Problem

0 xfxθ

A. θ=-20o

B. θ=-22o

70 m/s

40 m

�

vyf = !gt f = !28m/s

�

tan(!) =vyf

vxf="28

70

C. θ=-24o

D. θ=-26o

modified problemvi

y

xθ

Sandra manages to throw a spareset of keys into her open window 4.0m off the ground. She throws from5.0 m away at an angle of 60o. Atwhat speed did the keys leave herhand?

θ

vi

vxi=vi cos(60o) = vi/2

vyi= vi sin(60o)

1. Decompose vi 2. Analyze one component

A. x-motion for xf

C. x-motion for vxf

B. x-motion for t

D. y-motion for t

4 m

5 m

Which Equation would you use?

ABCD�

vxf = vxi + axt

�

x f = xi + vxit +1

2axt

2

�

x f = xi +1

2(vxf + vxi)t

�

vxf2

= vxi2

+ 2ax (x f ! xi)

modified problemvi

y

xθ

Sandra manages to throw a spareset of keys into her open window 4.0m off the ground. She throws from5.0 m away at an angle of 60o. Atwhat speed did the keys leave herhand?

θ

vi

vxi=vi cos(60o) = vi/2

vyi= vi sin(60o)

1. Decompose vi 2. Analyze one component

A. x-motion for xf

C. x-motion for vxf

B. x-motion for t

D. y-motion for t

4 m

5 m

�

x f = xi + vxit +1

2axt

2

5 = 0 + 0.5 vi tf + 0 tf = 10 m/vi

Which Equation would you substitutethe time expression into?

ABCD�

vyf = vyi + ayt

�

y f = yi + vyit +1

2ayt

2

�

y f = yi +1

2(vyf + vyi)t

�

vyf2

= vyi2

+ 2ay (y f ! yi)

modified problemvi

y

xθ

Sandra manages to throw a spareset of keys into her open window 4.0m off the ground. She throws from5.0 m away at an angle of 60o. Atwhat speed did the keys leave herhand?

θ

vi

vxi=vi cos(60o) = vi/2

vyi= vi sin(60o)

1. Decompose vi 3. Eliminate t from y-motion

4 m

5 m

tf = 10 m/vi

�

y f = yi + vyit !1

2gt

2

�

4 = 0 + vi sin(60o)10

vi!1

2g100

vi2

A. vi=5.2 m/sB. vi=8.6 m/s

C. vi=10.2 m/s

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wily monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wily monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

ymf ?

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wily monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

ymf ?

A. x-motion for tB. x-motion for vxfC. y-motion for tD. y-motion for vyf

First step is

Which Equation would you use?

ABCD�

vxf = vxi + axt

�

x f = xi + vxit +1

2axt

2

�

x f = xi +1

2(vxf + vxi)t

�

vxf2

= vxi2

+ 2ax (x f ! xi)

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wily monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

ymf ?

A. x-motion for tB. x-motion for vxfC. y-motion for tD. y-motion for vyf

First step is

�

xm = vi cos(!)t f

�

t f =xm

vi cos(!)

Which Equation would you substitutethe time expression into?

ABCD�

vyf = vyi + ayt

�

y f = yi + vyit +1

2ayt

2

�

y f = yi +1

2(vyf + vyi)t

�

vyf2

= vyi2

+ 2ay (y f ! yi)

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wily monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

ymf ?�

t f =xm

vi cos(!)

�

y f = vi sin(!)t f "1

2gt f

2

�

y f = xm tan(!) "1

2gt f

2

Monkey & HunterA monkey sits in a tree. A hunter having aimed directly at himfires a bullet at speed vi. At that precise instant the monkey dropsfrom the tree. Where does the bullet pass the wiley monkey? A. below B. above C. it hits the monkey

θx

y

ymi

xm

vi

ymf ?�

y f = xm tan(!) "1

2gt f

2

�

ymi = xm tan(!)

�

ymf = ymi !1

2gt f

2

�

ymf = xm tan(!) "1

2gt f

2

viy

xθ

What angle should the outfielderthrow the ball, with initial speed|vi|=30 m/s, to go half as far? (40 m)

�

x f =vi2

gsin(2!)

�

x f g

vi2

= sin(2!) =40m" 9.8ms

#2

(30ms#1)2

= 0.39

sin(2θ)=0.39 , θ=11o

viy

xθ

What angle should the outfielderthrow the ball, with initial speed|vi|=30 m/s, to go half as far? (40 m)

�

x f =vi2

gsin(2!)

�

x f g

vi2

= sin(2!) =40m" 9.8ms

#2

(30ms#1)2

= 0.39

sin(2θ)=0.39 , θ=11o

The ball bounces perfectly & repeats the trajectory.How long does it take to go 80.1 m this way?

�

t f =2visin(!)

g=1.2s

A. 1.2 s

C. 3.1 sD. 4.2 s

B. 2.4 s