Vectors and Projectile Motion

Chapter 3

Adding Vectors

When adding vectors that fall on the same line, using pluses and minuses is sufficient.

When dealing with two dimensions, vector addition must be used. A complete answer includes magnitude, units, and a direction usually indicated by an angle.

Right Angle Vector AdditionC=resultant, means sum of two vectors.

2m

4m

=4.47m

c

In this case, Pythagorean theorem will solve for c.

In order to find , trig must be used. It is usually best to use the given sides, rather than the calculated side.

=63.4o

If you walk 2m north and then 4m east, what is your displacement?

Steps for Adding Perpendicular Vectors

Line vectors up head to tail.

Draw the resultant.

Find the resultant length using the Pythagorean theorem.

Use trig to find the angle.

Ensure that the angle measure has a distinct reference point, either by words or a picture.

Practice Problem

5m/s

10m/s

A sailboat experiences a 5m/s wind to the east while traveling 10m/s downstream. Find the resulting net velocity

5m/s

10m/s

Adding more than two vectors

Vectors can be added in any order.

Subtracting a vector is the same as adding a vector that is the same length, but going the exact opposite direction.

When drawing a picture, vectors may be rearranged into any order as long as their orientation is maintained.

Example

Practice

What is the total displacement of the motion, assuming each block is 1m long?

Vector directionDegrees on the unit circle

Start on the positive x and move CCW for positive degrees and CW for negative degrees.Examples, 40m @ 217°, 34m/s @ -73°

Compass direction27° S of WSouthwest would indicate a 45°Earth and space sciences use a system where north is 0° and one always has a positive angle and proceeds CW around the compass.

Adding vectors with random orientation

Components-A set of vectors that have a sum equal to a given vector.

Usually most useful to have components perpendicular to each other.

Breaking up a vector, called resolution, is the exact opposite of adding two perpendicular vectors to find a resultant.

Usually, components are along the x and y axis.

x

y

Component practice

50m@200°32m@67°

14m/s@-112° 6m/s@-300°

Steps for solving vector problems

Graphically represent the system to give yourself a check for your answer.Draw each vector individually.Using trig, find the x and y components of each vector.Add all the x components together and add all the y components together.Make a new diagram using the total x component and the total y component. Draw in the resultant and label a reference angle.Use the Pythagorean theorem and trig to find the resultant and angle

Example Problem

9m, 45o up from the horizontal

11m, 18o down from the horizontal

Splitting the Triangles Up

y1

9m, 45o up from the horizontal

x1

45o

11m, 18o down from the horizontal

x2

y2

18o

y1

x1

45o

x2

y2

18o

Working with the first triangle

945sin 1y

h

o

945cos 1x

h

a

my 36.645sin9 1

mx 36.645cos9 1 9m, 45o up from the horizontal

x1

45o

y1

Working with the second triangle

my 4.318sin11 2

1118cos 2x

h

a

1118sin 2y

h

o

mx 4.1018cos11 2 11m, 18o down from

the horizontal

x2

y2

18o

Putting everything together

myyy

mxxx

tot

tot

96.2

8.16

21

21

8.16

96.2tan

tot

tot

x

y

mc

mc

bac

1.17

9.291

8.1696.22

22222

xtot

ytot

c

01 1.108.16

96.2tan

Practice

Find the resultant of the following vectors:

450m @ 20o

360m @ 300o

290m @ 189o

405m @ 115o

Projectile Motion

Horizontal and 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

Cliff Problems

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 AUse 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

Practice Problems

A ball is thrown horizontally from the roof of a building 56m tall and lands 45m away from the base. What was the ball’s initial speed?

An onion runs off of a building at a speed of 22.2m/s and lands 36.0m from the base of the building. How tall is the building?

Starting and Ending at the Same Height

g

vR i 2sin2

g

vy i

2

sin 22

max

g

vt i sin2

Practice Problem

A water balloon is shot from a slingshot at an angle of 21o and with a velocity of 16m/s. What is:

The horizontal range?

The time it’s in the air?

It’s maximum height?

More Practice

A long jumper leaves the ground at 30o and travels 7.8m. How long is he in the air?

A pilot drops a flaming bag of poo from a plane in an attempt to hit the roof of the high school. If the plane is traveling at 160km/hr and is 160m above the ground, how far before he is directly overhead should the pilot drop the bag?