Physics 111: Lecture 2 Todays Agenda

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Physics 111 Lecture 2Todays Agenda

• Recap of 1-D motion with constant acceleration
• 1-D free fall
• example
• Review of Vectors
• 3-D Kinematics
• Shoot the monkey
• Baseball
• Independence of x and y components

2
Review

• For constant acceleration we found

x
t
v

• From which we derived

t
a
t
3
Recall what you saw
4
1-D Free-Fall

• This is a nice example of constant acceleration
  (gravity)
• In this case, acceleration is caused by the force
  of gravity
• Usually pick y-axis upward
• Acceleration of gravity is down

y
t
v
t
a
y
t
ay ? g
5
Gravity facts
Ball w/ cup
Penny feather

• g does not depend on the nature of the material!
• Galileo (1564-1642) figured this out without
  fancy clocks rulers!
• demo - feather penny in vacuum
• Nominally, g 9.81 m/s2
• At the equator g 9.78 m/s2
• At the North pole g 9.83 m/s2
• More on gravity in a few lectures!

6
Problem

• The pilot of a hovering helicopter drops a lead
  brick from a height of 1000 m. How long does it
  take to reach the ground and how fast is it
  moving when it gets there? (neglect air
  resistance)

7
Problem

• First choose coordinate system.
• Origin and y-direction.
• Next write down position equation
• Realize that v0y 0.

1000 m
y 0
8
Problem

• Solve for time t when y 0 given that y0 1000
  m.
• Recall
• Solve for vy

y0 1000 m
y 0
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Lecture 2, Act 11D free fall

• Alice and Bill are standing at the top of a cliff
  of height H. Both throw a ball with initial
  speed v0, Alice straight down and Bill straight
  up. The speed of the balls when they hit the
  ground are vA and vB respectively. Which of the
  following is true
• (a) vA lt vB (b) vA vB (c) vA gt
  vB

v0
Bill
Alice
v0
H
vA
vB
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Lecture 2, Act 11D Free fall

• Since the motion up and back down is symmetric,
  intuition should tell you that v v0
• We can prove that your intuition is correct

Equation
This looks just like Bill threw the ball down
with speed v0, sothe speed at the bottom
shouldbe the same as Alices ball.
Bill
v0
v v0
H
y 0
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Lecture 2, Act 11D Free fall

• We can also just use the equation directly

Alice
same !!
Bill
v0
Alice
Bill
v0
y 0
12
Vectors (review)

• In 1 dimension, we could specify direction with a
  or - sign. For example, in the previous
  problem ay -g etc.
• In 2 or 3 dimensions, we need more than a sign to
  specify the direction of something
• To illustrate this, consider the position vector
  r in 2 dimensions.

• Example Where is Chicago?
• Choose origin at Urbana
• Choose coordinates of distance (miles), and
  direction (N,S,E,W)
• In this case r is a vector that points 120
  miles north.

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Vectors...

• There are two common ways of indicating that
  something is a vector quantity
• Boldface notation A
• Arrow notation

A
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Vectors...

• The components of r are its (x,y,z) coordinates
• r (rx ,ry ,rz ) (x,y,z)
• Consider this in 2-D (since its easier to draw)
• rx x r cos ???
• ry y r sin ???

where r r
(x,y)
y
????arctan( y / x )
r
?
x
15
Vectors...

• The magnitude (length) of r is found using the
  Pythagorean theorem

r
y
x

• The length of a vector clearly does not depend on
  its direction.

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Unit Vectors

• A Unit Vector is a vector having length 1 and no
  units
• It is used to specify a direction
• Unit vector u points in the direction of U
• Often denoted with a hat u û
• Useful examples are the Cartesian unit vectors
  i, j, k
• point in the direction of the x, y and z axes

U
y
j
x
i
k
z
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Vector addition

• Consider the vectors A and B. Find A B.

B
B
A
A
A
C A B
B

• We can arrange the vectors as we want, as long as
  we maintain their length and direction!!

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Vector addition using components

• Consider C A B.
• (a) C (Ax i Ay j) (Bx i By j) (Ax
  Bx)i (Ay By)j
• (b) C (Cx i Cy j)
• Comparing components of (a) and (b)
• Cx Ax Bx
• Cy Ay By

By
C
B
Bx
A
Ay
Ax
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Lecture 2, Act 2Vectors

• Vector A 0,2,1
• Vector B 3,0,2
• Vector C 1,-4,2

What is the resultant vector, D, from adding
ABC?
(a) 3,5,-1 (b) 4,-2,5 (c) 5,-2,4
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Lecture 2, Act 2Solution
D (AXi AYj AZk) (BXi BYj BZk) (CXi
CYj CZk) (AX BX CX)i (AY BY
CY)j (AZ BZ CZ)k (0 3 1)i (2 0
- 4)j (1 2 2)k 4,-2,5
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3-D Kinematics

• The position, velocity, and acceleration of a
  particle in 3 dimensions can be expressed as
• r x i y j z k
• v vx i vy j vz k (i , j , k
  unit vectors )
• a ax i ay j az k
• We have already seen the 1-D kinematics equations

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3-D Kinematics

• For 3-D, we simply apply the 1-D equations to
  each of the component equations.
• Which can be combined into the vector equations
• r r(t) v dr / dt a d2r / dt2

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3-D Kinematics

• So for constant acceleration we can integrate to
  get
• a const
• v v0 a t
• r r0 v0 t 1/2 a t2
• (where a, v, v0, r, r0, are all vectors)

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2-D Kinematics
lost marbles

• Most 3-D problems can be reduced to 2-D problems
  when acceleration is constant
• Choose y axis to be along direction of
  acceleration
• Choose x axis to be along the other direction
  of motion
• Example Throwing a baseball (neglecting air
  resistance)
• Acceleration is constant (gravity)
• Choose y axis up ay -g
• Choose x axis along the ground in the direction
  of the throw

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x and y components of motion are independent.
Cart

• A man on a train tosses a ball straight up in the
  air.
• View this from two reference frames

Reference frame on the moving train.
Reference frame on the ground.
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Problem

• Mark McGwire clobbers a fastball toward
  center-field. The ball is hit 1 m (yo ) above
  the plate, and its initial velocity is 36.5 m/s
  (v ) at an angle of 30o (?) above horizontal.
  The center-field wall is 113 m (D) from the plate
  and is 3 m (h) high.
• What time does the ball reach the fence?
• Does Mark get a home run?

v
h
?
y0
D
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Problem...

• Choose y axis up.
• Choose x axis along the ground in the direction
  of the hit.
• Choose the origin (0,0) to be at the plate.
• Say that the ball is hit at t 0, x x0 0
• Equations of motion are
• vx v0x vy v0y - gt
• x vxt y y0 v0y t - 1/ 2 gt2

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Problem...

• Use geometry to figure out v0x and v0y

g
Find v0x v cos ?. and v0y v sin ?.
y
v
v0y
?
y0
v0x
x
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Problem...

• The time to reach the wall is t D / vx
  (easy!)
• We have an equation that tell us y(t) y0 v0y
  t a t2/ 2
• So, were done....now we just plug in the
  numbers
• Find
• vx 36.5 cos(30) m/s 31.6 m/s
• vy 36.5 sin(30) m/s 18.25 m/s
• t (113 m) / (31.6 m/s) 3.58 s
• y(t) (1.0 m) (18.25 m/s)(3.58 s) -
  
  (0.5)(9.8 m/s2)(3.58 s)2
• (1.0 65.3 - 62.8) m 3.5 m
• Since the wall is 3 m high, Mark gets the homer!!

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Lecture 2, Act 3Motion in 2D

• Two footballs are thrown from the same point on a
  flat field. Both are thrown at an angle of 30o
  above the horizontal. Ball 2 has twice the
  initial speed of ball 1. If ball 1 is caught a
  distance D1 from the thrower, how far away from
  the thrower D2 will the receiver of ball 2 be
  when he catches it?(a) D2 2D1 (b) D2 4D1
  (c) D2 8D1

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Lecture 2, Act 3Solution

• The distance a ball will go is simply x
  (horizontal speed) x (time in air) v0x t

• To figure out time in air, consider the
  equation for the height of the ball

• When the ball is caught, y y0

(time of catch)
(time of throw)
two solutions
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Lecture 2, Act 3Solution
x v0x t

• So the time spent in the air is proportional to
  v0y

• Since the angles are the same, both v0y and v0x
  for ball 2are twice those of ball 1.

• Ball 2 is in the air twice as long as ball 1, but
  it also has twice the horizontal speed, so it
  will go 4 times as far!!

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Shooting the Monkey(tranquilizer gun)

• Where does the zookeeper aim if he wants to hit
  the monkey?
• ( He knows the monkey willlet go as soon as he
  shoots ! )

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Shooting the Monkey...

r r0

• If there were no gravity, simply aim

at the monkey
r v0t
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Shooting the Monkey...
r r0 - 1/2 g t2

• With gravity, still aim at the monkey!

r v0 t - 1/2 g t2
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RecapShooting the monkey...
x v0 t y -1/2 g t2

• This may be easier to think about.
• Its exactly the same idea!!

x x0 y -1/2 g t2
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Recap of Lecture 2

• Recap of 1-D motion with constant acceleration.
  (Text 2-3)
• 1-D Free-Fall (Text 2-3)
• example
• Review of Vectors (Text 3-1 3-2)
• 3-D Kinematics (Text 3-3 3-4)
• Shoot the monkey (Ex. 3-11)
• Baseball problem
• Independence of x and y components
• Look at textbook problems Chapter 3 35, 39,
  69, 97