Vectors & Projectile Motion

Chapter 3

Horizontal & Vertical Motion

Things to remember for 2D motion

• Vertical and horizontal motion are independent.• When something is traveling through the air, ignore

the effects of air resistance.• There is nothing pushing or pulling a projectile

horizontally, therefore ax=0.• For vertical motion, gravity is causing the vertical

acceleration, so ay=-9.8m/s2.• We will assume that projectiles landing at a height

different from their initial height are always launched horizontally. Therefore, any initial velocity is an x piece. There is no y component for initial velocity.

Horizontal Components

tvx

tvtx

tvatx

ix

ix

ix

2

2

021

21

ixfx

ixfx

ixfx

vv

xvv

axvv

02

222

22

ixfx

ixfx

ixfx

vv

tvv

atvv

0

tvx

tvx

tvvx

tvvx

ix

ix

ixix

fxix

221

21

21

a=0

Vertical Components

2

2

2

21

021

21

gty

tgty

tvaty iy

gyv

gyv

ayvv

fy

fy

ify

2

20

22

22

gtv

gtv

atvv

fy

fy

ify

0

tvy

tvy

tvvy

f

fy

fyiy

21

021

21

Viy=0

a=g

Example #7 – A Cliff Problem

• A car drives off a 100m cliff at a speed of 47m/s. What is:

– The time it takes to hit the ground?

– It’s horizontal distance from the base of the cliff?

– It’s final velocity?

Identify Vertical and Horizontal Components

Horizontal

• vi=47m/s

• vf=47m/s

• a=0• t=?• x=?

Vertical• a=g=-9.8m/s2

• vi=0

• y=100m• t=?

• vf=?

Part A

• Use y pieces because there is not enough information to solve for t using x components.

• Since the car stops moving horizontally at the same time it stops moving vertically, the t found using the y components can be used for the x components as well.

tvaty iy 2

21

stg

y5.4

2

22t

a

y

Part B

tvatx i 2

21

mtvx i 3.212

Part C

• The car is going both down and over at the end so vf has both x and y components.

• vfx= vix

• vfy must be calculated.

atvv if

sm

f atv 3.44

Part C Continued

sm

fyfx

c

cvv

6.64

222

01 3.43tan

tan

fx

fy

fx

fy

v

v

v

v

a

o

• Put x and y components together and solve for both resultant and angle.

c

vfx

vfy

Example #8

A ball is launched horizontally at 10 m/sec off of a 1.2 m tall table. Calculate the following:A) The time in the air.B) How far from the table does

the ball land?C) What is the velocity of the ball

just before it hits the ground?

Example #9

A ball is launched off of a table. If it lands 3.4 m away from 0.85 m tall table, A) How long is it in the air?

B) What is the launch velocity?

C) What is the final velocity of the ball?

Projectile Motion in Football

Starting and Ending at the Same Height

g

vR i 2sin2

g

vy i

2

sin 22

max

g

vt i sin2

Example #10A cannon ball gets fired at a 30 degree angle at 26 m/sec.A) How far away does it land?

B) How long is it in the air?

C) What is the maximum height?

D) How fast is it moving when it hits the ground?

E) How fast is the ball moving and how high is the ball 0.6 sec after launch?

Example #11

A) How fast is it hit?

B) What is the velocity just before the ball hits the ground?

C) How high did it go?

D) Some lady next to him hit the ball with the same velocity but instead the ball went very high. Her ball landed at the same place. At what angle did she hit the ball?

A golf ball is driven at a driving range and lands 100 m away from the golfer. If it is hit at a 15 degree angle,

Monkey Hunter!