PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 3

• HW 2 due: Wednesday Jan 23 @ 3:59 am

• (MLK Jr. Day on Jan 21)

• Note: related reading for each lecture listed onCalendar page at PHY 231 website

Announcement

Main points of last lecture

• Acceleration defined:

• Equations with constantAcceleration:

(Δx, v0, vf, a, t)

• Acceleration of freefall:

a =v f ! vi

tbasic equations:

1) v = v0 + at

2) !x =1

2(v0 + v)t

3) !x = v0t +1

2at2

4) !x = v f t "1

2at2

5) a!x =v f2

2"v02

2

!

a" (#g) = #9.81m/s2

Example 2.9a

A man throws a brick upward from thetop of a building. (Assume the coordinatesystem is defined with positive definedas upward)

At what point is the acceleration zero?

A C

D

A C

D

B

E

a) Ab) Bc) Cd) De) None of the above

Example 2.9b

A man throws a brick upward from thetop of a building. (Assume the coordinatesystem is defined with positive definedas upward)

At what point is the velocity zero?

A C

D

A C

D

B

E

a) Ab) Bc) Cd) De) None of the above

CHAPTER 3

Two-Dimensional Motion andTwo-Dimensional Motion andVectorsVectors

Scalars and Vectors

• Scalars: Magnitude only

• Examples: time, distance, speed,…

• Vectors: Magnitude and Direction

• Examples: displacement, velocity, acceleration,…

Representations:Representations:

x

y

(x, y)

(x, y) (r, θ)

Vectors in 2 Dimensions

Vector distinguished byarrow overhead: A

Cartesian Polar

Vector Addition/Subtraction

• 2nd vector begins at end offirst vector

• Order doesn’t matter

Vector addition

Vector subtraction

A – B can be interpretedas A+(-B)• Order does matter

Vector Components

Cartesian components areprojections along the x-and y-axes

Ax= Acos!

Ay = Asin!

Going backwards,

A = Ax2+ Ay

2and ! = tan"1

Ay

Ax

Example 3.1a

The magnitude of (A-B) is :

a) <0b) =0c) >0

Example 3.1b

The x-component of (A-B) is:

a) <0b) =0c) >0

Example 3.1c

The y-component of (A-B) > 0

a) <0b) =0c) >0

Example 3.2

Some hikers walk due east from the trail headfor 5 miles. Then the trail turns sharply tothe southwest, and they continue for 2 moremiles until they reach a waterfalls. What isthe magnitude and direction of thedisplacement from the start of the trail to thewaterfalls?

3.85 miles, at -21.5 degrees

5 mi

2 mi

2-dim Motion: Velocity

Graphically,

v = Δr / Δt

It is a vector(rate of change of position)

Trajectory

Multiplying/Dividing Vectors by Scalars

• Example: v = Δr / Δt

• Vector multiplied by scalar is a vector: B = 2A

• Magnitude changes proportionately: |B| = 2|A|

• Direction is unchanged: θB = θA

B

A

2-d Motion with constant acceleration

• X- and Y-motion are independent

• Two separate 1-d problems:• Δx, vx, ax• Δy, vy, ay

• Connected by time t

• Important special case: Projectile motion• ax=0 • ay=-g

Projectile Motion

• X-direction: (ax=0)

• Y-direction: (ay=-g)

Note: we ignore• air resistance• rotation of earth

!

vx

= constant

"x = vxt

!

vy, f = vy,0 " gt

#y = 1

2(vy,0 + vy, f )t

#y = vy,0t "1

2gt

2

#y = vy, f t + 1

2gt

2

"g#y =vy, f2

2"vy,02

2

Projectile Motion

Accelerationis constant

Pop and Drop Demo

The Ballistic CartDemo

Finding Trajectory, y(x)1. Write down x(t)

2. Write down y(t)

3. Invert x(t) to find t(x)

4. Insert t(x) into y(t) to get y(x)

Trajectory is parabolic

x = v0,xt

y = v0,yt !1

2gt2

t = x / v0,x

y =v0,y

v0,xx !

1

2

g

v0,x2x2

Example 3.3

An airplane drops food totwo starving hunters. Theplane is flying at an altitudeof 100 m and with a velocityof 40.0 m/s.

How far ahead of thehunters should the planerelease the food?

X181 m

h

v0

Example 3.4a

h

Dθ

v0

The Y-component of v at A is :a) <0b) 0c) >0

Example 3.4b

h

Dθ

v0

a) <0b) 0c) >0

The Y-component of v at B is

Example 3.4c

h

Dθ

v0

a) <0b) 0c) >0

The Y-component of v at C is:

Example 3.4d

h

Dθ

v0

a) Ab) Bc) Cd) Equal at all points

The speed is greatest at:

Example 3.4e

h

Dθ

v0

a) Ab) Bc) Cd) Equal at all points

The X-component of v is greatest at:

Example 3.4f

h

Dθ

v0

a) Ab) Bc) Cd) Equal at all points

The magnitude of the acceleration is greatest at:

Range Formula

• Good for when yf = yi

x = vi,xt

y = vi,yt !1

2gt2= 0

t =2vi,y

g

x =2vi,xvi,y

g=2vi

2cos" sin"

g

x =vi2

gsin2"

Range Formula

• Maximum for θ=45°R =vi2

gsin2!

Example 3.5a

100 m

A softball leaves a bat with aninitial velocity of 31.33 m/s. Whatis the maximum distance one couldexpect the ball to travel?

Example 3.6

68 m

A cannon hurls a projectile which hits a target located on acliff D=500 m away in the horizontal direction. The cannonis pointed 50 degrees above the horizontal and the muzzlevelocity is 75 m/s. Find the height h of the cliff?

h

Dθ

v0