Motion in Two and Three Dimensions

Distance• Distance - How far you actually traveled.• Displacement - Change in your position.

– This is a vector and direction is important. Ex. You travel from Dayton, to Indianapolis, to

Columbus. When your trip is done, what is your distance traveled and your displacement?

170 km 105 km

ColumbusIndianapolis Dayton

Position Vector• Scalar - Magnitude (Size)• Vector - Magnitude and Direction• Position Vector - Location relative to an

origin.

3.5 m

25°

x

y

Vector Addition• Place vectors tail to head.• Sum is from the tail of the first to the head

of the last vector.

A

B

A B

CSolve

Graphically 700.7 mA

750.5 mB

261.4 mC

BAC

Vector Subtraction• Same as adding a negative.• -1 changes the vector’s direction by 180°.

700.7 mA

750.5 mB

261.4 mC

ACB

A

C

A

B

C

SolveGraphically

-A

Additional Properties

• Multiplication of a vector by a scalar– Can change the length of the vector.– Can change the sign of the vector.

• Algebraic Properties of Vectors– Commutative– Associative– Distributive

Law of Sines & Cosines

• Can perform vector addition using the laws of sines and cosines.

A B

C

700.7 mA

750.5 mB

261.4 mC

7 535°

Law of Cosines Law of Sines

cos2²²² ABBAC CBA sinsinsin

Coordinate Systems• Project the vector on to the axis of the

coordinate system.• Ordered Pair of coordinates

AAY

AX

YX AAA ,

cosAAX sinAAY

22YX AAA

X

Y

AA1tan

• Convert back to polar

Unit Vectors• Chose a vector of length one in the direction

of each axis of the coordinate system.

0,1ˆ i 1,0ˆ jj

i x

y

YX AAA ,

• Ordered Pair becomes

jAiAA YXˆˆ

Vector Addition (Again)

• Break each vector into components.

A BC

Ax

Ay

Bx

By

CxCy

700.7 mA

750.5 mB mjiA ˆ6.6ˆ4.2

mjiB ˆ8.4ˆ3.1

• Add each set of components together.

261.4 mC

mjiC ˆ8.1ˆ7.3

BAC

• A tracking station picks up the Aurora at a location

3.1 seconds later it is located at

What is the magnitude of the displacement?

What is its average velocity?

The Aurora

mkjir }ˆ100ˆ1100ˆ1800{1

mkjir }ˆ500ˆ1500ˆ3500{2

Ferris Wheel You are located on a moving Ferris

Wheel at King’s Island. Which of the following describes your motion.A) You are stationary.B) You are moving in a straight line.C) You are moving in a circle.D) You are moving in little loops

around a larger circle.

Position, Velocity, Acceleration

• Position

• Velocity

• Acceleration

trvavg

dtrd

trv

t

0

lim

if rrr

tv

ttvv

aif

ifavg

dtvd

tva

t

0

lim

Instantaneous

Acceleration and Velocity

• Constant Acceleration

• Ex. A rocket is traveling at a velocity of when its engines fail.

What is its velocity after 20 s?

dtvda

fv

v

t

vddta

00

tavv f

0

smki / ˆ250ˆ55

Acceleration and Displacement

• Integrating again gives

• Ex. A rocket is traveling at a velocity of when its engines fail.

What is its displacement after 20s?

221

00 tatvrrf

smki / ˆ250ˆ55

Galilean Transform• Galilean transform is used when comparing

velocities between two reference frames. (At least one is moving.)

x’

y’

x

y v v’

V O’O

O - Stationary FrameO’ - Moving Frame

Vvv

'

P

Vvv

'

or

At Sea• Ex. A ship leaves Miami traveling due east

at 6.00 m/s. It crosses the Gulf Stream, which is running at 1.79 m/s 75°. In what direction and at what speed does the ship travel with respect to Miami?

2-Dimensional Problems

• When solving 2-D problems, how many variables can there be?

Initial vertical positionInitial horizontal positionInitial speedInitial angle of speedHorizontal accelerationVertical accelerationInitial time

Final vertical positionFinal horizontal positionFinal speedFinal angle of motionFinal time

What is the minimum you need?

Projectile Motion• Initial Velocity• Acceleration• X-component of

displacement (x0=0)• Y-component of

displacement (y0=0)

jvivv iiiiiˆsinˆcos

2/ ˆ8.9ˆ smjjga

tvx iif cos2

21 sin gttvy iif

Catapult• A catapult is located in a castle that is 50m

above the surrounding terrain. At what velocity must the catapult launch an object in order to hit a location 860m away if the launch angle is 50°? v0

0

Centripetal Acceleration

• Centripetal Acceleration - Object moves in a circle at constant speed.

• Acceleration velocity

vf

v0r0

rf

r

v0vf

vrr

vv

rtv

vv

rv

tva

2

Rounding the Curve• While driving at 24.5 m/s, you round a turn

with a radius of curvature of 120 m. What is your acceleration? What direction is it?

r

v

Non-Uniform Circular Motion

• Radial Acceleration - Perpendicular to the velocity. Radially inward towards the center of the circular path.

• Tangential Acceleration - Parallel to the velocity. Slowing down and speeding up.

ar

at

ar

at

SpeedingUp

SlowingDown

Ball on a String• Each dot represents the position of a

spinning object in equal time intervals. Indicate the acceleration at each dot.

Cedarville 500

A

B

C D

EF

G

HI

J

KLM

Identify tangential acceleration & decelerationIdentify zero & high radial acceleration

Under Siege• A cannon is fired at an angle of 55° with a

muzzle velocity of 300 m/s. The shell hits a castle which is at a 100m higher elevation. How long do the residents have to take cover?

xf

v0

h