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

Going backwards,
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Example 3.1
a) F b) F c) T
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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 10 miles traveling 30
degrees north of east. Carol, who is bringing the
glasses, takes a short cut which 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
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 magnitudeo the
  orginial vector multiplied/divided by the
  scalar
• Direction of new vector is the same or opposite
  to original vector

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Principles of 2-d Motion

• X- and Y-motion are independent
• Can be treated as two separate 1-d problems
• To get trajectory (x vs. y)
• 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)
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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 3.4

1. The Y-component of v at A is (lt0, 0, gt0)
2. The Y-component of v at B is (lt0, 0, gt0)
3. The Y-component of v at C is (lt0, 0, gt0)
4. The total velocity is greatest at (A,B,C)
5. The X-component of v is greatest at (A,B,C)

1.gt02. 03. lt04. A5. 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 3.5b
A cannon aims a projectile at a target located on
a cliff 500 m away in the horizontal direction
and 75 meters above the cannon. The cannon is
pointed 50 degrees to the horizontal. What muzzle
velocity should the cannon employ to hit the
target?
75.4 m/s
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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)25.6?
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
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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 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 capable of moving 200 mph in still
air. A wind blows directly from the North at 50
mph. If the airplane accounts for the wind (by
pointing the plane somewhat into the wind) and
flies directly east relative to the ground.
What is his resulting ground speed? In what
direction is the nose of the plane pointed?
193.6 mph 14.5 deg. north of east