PHYSICS 231

INTRODUCTORY PHYSICS I

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 on Calendar page at PHY 231 website

AnnouncementAnnouncement

Main points of last lecture

• Acceleration defined:

• Equations with constantAcceleration:

(x, v0, vf, a, t)

• Acceleration of freefall:

a =vf −vit

basic equations:

1) v =v0 +at

2) x=12(v0 +v)t

3) x=v0t+12at2

4) x=vft−12at2

5) ax=vf2

2−v02

2

€

a → (−g) = −9.81m/s2

Example 2.9a

A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positive defined as 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 the top of a building. (Assume the coordinate system is defined with positive defined as 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 Two-Dimensional Motion and Vectorsand Vectors

Two-Dimensional Motion Two-Dimensional Motion and Vectorsand Vectors

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, )

Vectors in 2 Dimensions

Vector distinguished byarrow overhead: A

Cartesian Polar

Vector Addition/Subtraction

• 2nd vector begins at end of first vector

• Order doesn’t matter

Vector addition

Vector subtraction

A – B can be interpreted as A+(-B)

• Order does matter

Vector Components

Cartesian components are projections along the x- and y-axes

Ax =Acosθ

Ay = Asinθ

Going backwards,

A = Ax2 + Ay

2 and =tan−1AyAx

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 head for 5 miles. Then the trail turns sharply to the southwest, and they continue for 2 more miles until they reach a waterfalls. What is the magnitude and direction of the displacement from the start of the trail to the waterfalls?

3.85 miles, at -21.5 degrees

5 mi

2 mi

2-dim Motion: Velocity

Graphically,

v = r / t

It is a vector(rate of change of position)

Trajectory

Multiplying/Dividing Vectors by Scalars

• Example: v = r / t

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

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

• Direction is unchanged: B = A

B

A

2-d Motion with constant acceleration

• X- and Y-motion are independent

• Two separate 1-d problems: x, vx, ax

y, 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

Δx = vxt

€

vy, f = vy,0 − gt

Δy = 12 (vy,0 + vy, f )t

Δy = vy,0t − 12 gt 2

Δy = vy, f t + 12 gt 2

−gΔy =vy, f

2

2−

vy,02

2

Projectile Motion

Acceleration is constant

Pop and Drop Demo

The Ballistic Cart Demo

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−12gt2

t =x/ v0,x

y =v0,yv0,x

x−12

gv0,x2 x2

Example 3.3

An airplane drops food to two starving hunters. The plane is flying at an altitude of 100 m and with a velocity of 40.0 m/s.

How far ahead of the hunters should the plane release the food?

X181 m

h

v0

Example 3.4a

h

D

v0

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

Example 3.4b

h

D

v0

a) <0b) 0c) >0

The Y-component of v at B is

Example 3.4c

h

D

v0

a) <0b) 0c) >0

The Y-component of v at C is:

Example 3.4d

h

D

v0

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

The speed is greatest at:

Example 3.4e

h

D

v0

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

The X-component of v is greatest at:

Example 3.4f

h

D

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−12gt2 =0

t=2vi,yg

x=2vi,xvi,y

g=2vi

2 cos sing

x=vi2

gsin2

Range Formula

• Maximum for =45R =vi2

gsin2

Example 3.5a

100 m

A softball leaves a bat with an initial velocity of 31.33 m/s. What is the maximum distance one could expect the ball to travel?

Example 3.6

68 m

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

h

D

v0