Two-Dimensional Motion and Vectors

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CHAPTER 3
Two-Dimensional Motion and Vectors
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VECTOR quantities
Vectors have magnitude and direction.
Representations
(x, y)
(r, q)
Other vectors velocity, acceleration, momentum,
force
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Vector Addition/Subtraction

• 2nd vector begins at end of first vector
• Order doesnt matter

Vector addition
A B can be interpreted as A(-B)
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Vector Components

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

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Example 3.1a
The magnitude of (A-B) is
a) lt0 b) 0 c) gt0
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Example 3.1b
The x-component of (A-B) is
a) lt0 b) 0 c) gt0
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Example 3.1c
The y-component of (A-B) gt 0
a) lt0 b) 0 c) gt0
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Example 3.2
Alice and Bob carry a bottle of wine to a picnic
site. Alice carries the bottle 5 miles due east,
and Bob carries the bottle another 10 miles
traveling 30 degrees north of east. Carol, who is
bringing the glasses, takes a short cut and goes
directly to the picnic site. How far did Carol
walk? What was Carols direction?
14.55 miles, at 20.10 degrees
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Arcsin, Arccos and Arctan Watch out!
samecosine
sametangent
same sine
Arcsin, Arccos and Arctan functions can yield
wrong angles if x or y are negative.
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2-dim Motion Velocity
v Dr / Dt It is a vector(rate of change of
position)
Graphically,
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Multiplying/Dividing Vectors by Scalars, e.g.
Dr/Dt

• Vector multiplied/divided by scalar is a vector
• Magnitude of new vector is magnitude of orginal
  vector multiplied/divided by scalar
• Direction of new vector same as original vector

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Principles of 2-d Motion

• X- and Y-motion are independent
• Two separate 1-d problems
• To get trajectory (y vs. x)
• Solve for x(t) and y(t)
• Invert one Eq. to get t(x)
• Insert t(x) into y(t) to get y(x)

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Projectile Motion

• X-motion is at constant velocity ax0,
  vxconstant
• Y-motion is at constant accelerationay-g

• Note we have ignored
• air resistance
• rotation of earth (Coriolis force)

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Projectile Motion
Acceleration is constant
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Pop and Drop Demo
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The Ballistic Cart Demo
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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
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Example 3.3
v0
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?
h
X
181 m
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Example 3.4a
The Y-component of v at A is
a) lt0 b) 0 c) gt0
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Example 3.4b
The Y-component of v at B is
a) lt0 b) 0 c) gt0
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Example 3.4c
The Y-component of v at C is
a) lt0 b) 0 c) gt0
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Example 3.4d
The speed is greatest at
a) A b) B c) C d) Equal at all points
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Example 3.4e
The X-component of v is greatest at
a) A b) B c) C d) Equal at all points
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Example 3.4f
The magnitude of the acceleration is greatest at
a) A b) B c) C d) Equal at all points
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Range Formula

• Good for when yf yi

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Range Formula

• Maximum for q45?

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Example 3.5a
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?
100 m
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Example 3.6
A cannon hurls a projectile which hits a target
located on a cliff D500 m away in the horizontal
direction. The cannon is pointed 50 degrees above
the horizontal and the muzzle velocity is 100
m/s. Find the height h of the cliff?
299 m
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Example 3.7, Shoot the Monkey
A hunter is a distance L 40 m from a tree in
which a monkey is perched a height h20 m above
the hunter. The hunter shoots an arrow at the
monkey. However, this is a smart monkey who lets
go of the branch the instant he sees the hunter
release the arrow. The initial velocity of the
arrow is v 50 m/s.
A. If the arrow traveled with infinite speed on a
straight line trajectory, at what angle should
the hunter aim the arrow relative to the ground?
qArctan(h/L)26.6?
B. Considering the effects of gravity, at what
angle should the hunter aim the arrow relative to
the ground?
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Solution
Must find v0,y/vx in terms of h and L
1. Height of arrow
2. Height of monkey
3. Require monkey and arrow to be at same place
Aim directly at Monkey!
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Shoot the Monkey Demo
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Relative velocity

• Velocity always defined relative to reference
  frame.All velocities are relative
• Relative velocities are calculated by vector
  addition/subtraction.
• Acceleration is independent of reference frame
• For high v c, rules are more complicated
  (Einstein)

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Example 3.8
A plane that is capable of traveling 200 m.p.h.
flies 100 miles into a 50 m.p.h. wind, then flies
back with a 50 m.p.h. tail wind. How long does
the trip take? What is the average speed of the
plane for thetrip?
1.067 hours 1 hr. and 4 minutes 187.4 mph
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Relative velocity in 2-d

• Sum velocities as vectors
• velocity relative to ground velocity relative
  to medium velocity of medium.

vbe vbr vre
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2 Cases
pointed perpendicularto stream
travels perpendicularto stream
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Example 3.9
An airplane is capable of moving 200 mph in still
air. The plane points directly east, but a 50 mph
wind from the north distorts his course. What is
the resulting ground speed? What direction does
the plane fly relative to the ground?
206.2 mph 14.0 deg. south of east
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Example 3.10
An airplane is capable of moving 200 mph in still
air. A wind blows directly from the North at 50
mph. The airplane accounts for the wind (by
pointing the plane somewhat into the wind) and
flies directly east relative to the ground.
What is the planes resulting ground speed? In
what direction is the nose of the plane pointed?
193.6 mph 14.5 deg. north of east