Lecture 10: Vectors, Projectile Motion

Please sit in the first six rows

A B C D F0

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Exam I Average = 83

Vector Components

y

x

A

Ax

Ay

θ

cossin

x

y

A AA A

y

xAx < 0

Ay

θ

φ

coscossin

x

y

A AA

A A

Adding vectors numerically

x x x

y y y

C A BC A BC A B

Also called algebraic additionWhat would subtraction look like?Scalar multiplication?

A

B

300

200

A = 15 mB = 25 m

Find Ax, Ay, Bx, By

, , ,x y C

C A BC C C

Examples

Slide 3-27

The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp?

The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp?

The vector has length 4.0 units.What are the x- and y-components of this

vector?

3.5, 2.0

2.0, 3.5

-3.5, 2

.0

2.0, -3.5

-3.5, -2

.0

20% 20% 20%20%20%

The vector has length 4.0 units.x- and y-components ?

1. 3.5, 2.02. 2.0, 3.53. -3.5, 2.04. 2.0, -3.55. -3.5, -2.0

Velocity as a vector

, same direction as

( , )

( )

( )

x

y

d dv dt t

dr x ydtdx tvdtdy tvdt

x

y1

2

Particle moves from 1 to 2 in time Δt. Draw the velocity vector.

Acceleration as a vector

,

f i

f i

yxx y

v v v dvat t t dt

dvdva adt dt

Note that unless object is moving in a straight line, a is not in the direction of v

Slide 3-28

The diagram below shows two successive positions of a particle; it’s a segment of a full motion diagram. Which of the acceleration vectors best represents the acceleration between vi and vf?

Motion on a Ramp

In the Soapbox Derby, young participants build cars with very low-friction wheels in which they roll down a hill. Cars racing on the track at Akron’s Derby Downs, where the national championship is held, begin on a 55 ft section of the track that is tipped 13° from the horizontal. What is the maximum possible acceleration of a car moving down this stretch of track? If a car starts from rest and accelerates at this rate for the full 55 ft, how fast will it be moving?

Slide 3-30

Projectile Motion

• Projectile launched at an angle θ above the horizontal with an initial velocity of vi.

• vix , viyy

x

vi

θ

Projectile motionx and y motion independent

2

0 a

( ) ( ) *

1( ) ( )2

x y

x ix y iy

i ix i iy

a g

v t v v t v g t

x t x v t y t y v gt

Problem 3:28A ball with a horizontal speed of 1.25 m/s rolls off a bench 1.00m above the floor.a) How long will it take the ball to hit the floor?b) How far from a point directly below the edge of

the bench will the ball hit the floor?

Circular Motion

There is an acceleration because the velocity is changing direction.

Slide 3-35

Circular Motion

Old vinyl records are 12" in diameter, and spin at 33⅓ rpm when played. What’s the acceleration of a point on the edge of the record?

Two friends are comparing the acceleration of their vehicles. Josh owns a Ford Mustang, which he clocks as doing 0 to 60 mph in a time of 5.6 seconds. Josie has a Mini Cooper that she claims is capable of a higher acceleration. When Josh laughs at her, she proceeds to drive her car in a tight circle at 13 mph. Which car experiences a higher acceleration?

Slide 3-36

Homework for TodayReading: 3: 1-6Problems: 3: CQ1, CQ3, MC19, MC20, 1, 2, 6, 7, 9, 12

MONDAY

Reading: 3: 7-8Problems: 3: CQ10, MC18, 13, 14, 16, 18, 21, 23, 26, 28, 30, 33