physics 231 introductory physics i · • 2nd vector begins at end of first vector • order...
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PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 3
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• 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
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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
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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
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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
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CHAPTER 3
Two-Dimensional Motion andTwo-Dimensional Motion andVectorsVectors
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Scalars and Vectors
• Scalars: Magnitude only
• Examples: time, distance, speed,…
• Vectors: Magnitude and Direction
• Examples: displacement, velocity, acceleration,…
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Representations:Representations:
x
y
(x, y)
(x, y) (r, θ)
Vectors in 2 Dimensions
Vector distinguished byarrow overhead: A
Cartesian Polar
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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
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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
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Example 3.1a
The magnitude of (A-B) is :
a) <0b) =0c) >0
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Example 3.1b
The x-component of (A-B) is:
a) <0b) =0c) >0
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Example 3.1c
The y-component of (A-B) > 0
a) <0b) =0c) >0
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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
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2-dim Motion: Velocity
Graphically,
v = Δr / Δt
It is a vector(rate of change of position)
Trajectory
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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
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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
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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
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Projectile Motion
Accelerationis constant
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Pop and Drop Demo
The Ballistic CartDemo
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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
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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
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Example 3.4a
h
Dθ
v0
The Y-component of v at A is :a) <0b) 0c) >0
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Example 3.4b
h
Dθ
v0
a) <0b) 0c) >0
The Y-component of v at B is
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Example 3.4c
h
Dθ
v0
a) <0b) 0c) >0
The Y-component of v at C is:
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Example 3.4d
h
Dθ
v0
a) Ab) Bc) Cd) Equal at all points
The speed is greatest at:
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Example 3.4e
h
Dθ
v0
a) Ab) Bc) Cd) Equal at all points
The X-component of v is greatest at:
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Example 3.4f
h
Dθ
v0
a) Ab) Bc) Cd) Equal at all points
The magnitude of the acceleration is greatest at:
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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"
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Range Formula
• Maximum for θ=45°R =vi2
gsin2!
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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?
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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