Warm-Up 09/02/10

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Warm-Up 09/02/10

• Vera is speeding down the interstate at 45.0 m/s
  when she sees an accident in the middle of the
  road. By the time Vera slams on the breaks, she
  is 50.0 m from the pileup and slows down at a
  rate of -10.0 m/s2. 1) Construct a
  velocity-time plot for Vera Side's motion. Use
  the plot to determine the distance which Vera
  would travel prior to reaching a complete stop
  (if she did not collide with the pileup). 2) Use
  kinematic equations to determine the distance
  which Vera Side would travel prior to reaching a
  complete stop (if she did not collide with the
  pileup). 3) Will Vera hit the cars in the pileup?
  

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• The distance traveled can be found by a
  calculation of the area between the line on the
  graph and the time axis.
• Area ½ bh 0.5(4.5 s)(45.0 m/s)
• Area 101 m

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• Given
• vi 45.0 m/s vf 0.0 m/s a -10.0 m/s2
• Find d (xf -xi) ??
• vf2 vi2 2a (xf -xi)
• (0 m/s)2 (45.0 m/s)2 2(-10.0 m/s2)d
• (-2025.0 m2/s2)/(-20.0 m/s2) d
• 101 m d
• Since the accident pileup is less than 101 m
  from Vera, she will indeed hit the pileup before
  completely stopping (unless she veers aside).

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Components of Motion

1. Motion in two dimensions, or curvilinear motion,
   is motion in which an object moves in a plane
   that can be described by a rectangular coordinate
   system.
2. Motion can be resolved, or analyzed, into
   rectangular components.

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An object in motion on a plane can be located
using two numbersthe x and y coordinates of its
position. Similarly, its velocity can be
described using components along the x- and
y-axes.
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For the ball rolling along the table
The velocity components are
The magnitude of the velocity vector is
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Vector Components Can describe displacement,
velocity, acceleration, and any other vector
quantity.
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Example 1

• An airplane is moving at a velocity of 250 mi/h
  in a direction 35o north of east. Find the
  components of the planes velocity in the
  eastward and northward direction.

Given v 250 mi/h q 35o Find Veast and Vnorth
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SOLUTION Veast V cos q (250 mi/h)cos35o
205 mi/h Vnorth V sin q (250 mi/h)sin35o
143 mi/h
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Displacement Components and Kinematic
Calculations
The components of the displacement are then given
by
Note that the x- and y-components are calculated
separately.
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More Equations of Motion
The equations of motion are
When solving two-dimensional motion problems,
each component, x and y, is treated separately.
The time is common to both.
http//www.physicsclassroom.com/mmedia/vectors/pla
ne.cfm
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If a motor boat were to head straight across a
river (that is, if the boat were to point its bow
straight towards the other side), it would not
reach the shore directly across from its starting
point. The river current influences the motion of
the boat and carries it downstream.
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Example 2

• A boat travels with a speed of 5.0 m/s in a
  straight path on a still lake. Suddenly, a steady
  wind pushed the boat perpendicular to its
  straight-line path with a speed of 3.0 m/s for
  5.0 s. Relative to the position just when the
  wind started to blow, where is the boat at the
  end of this time?
• Step One Sketch the Problem

? 3 m/s ?? 5 m/s
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• Solution Choose the x-axis as the original
  direction of the boat and y-axis as the direction
  of the wind.
• Given
• Vxo 5.0 m/s ax 0 m/s2 Vyo 3.0 m/s
• t 5.0 s ay 0 m/s2
• Find x and y
• x x0 Vxo t
• y y0 Vyo t

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Solve for x and y

• x 0 (5.0 m/s)(5s) 25 m
• y 0 (3.0 m/s)(5s) 15 m
• d sqrt(x2 y2) sqrt((25m)2(15m)2) 29m
• And
• tanq y/x
• tan-1(y/x) tan-1(15m/25m) 31o
• So the boat is now 29 m from where it started,
  at 31o to the horizontal.

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Curvilinear Motion
If the acceleration is not parallel to the
velocity, the object will move in a curve
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The diagram below shows the trajectory for a
projectile launched non-horizontally from an
elevated position on top of a cliff. The initial
horizontal and vertical components of the
velocity are 8 m/s and 19.6 m/s respectively.
Positions of the object at 1-second intervals are
shown. Determine the horizontal and vertical
velocities at each instant shown in the diagram.
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Physlet Exploration 3.1 Addition of Displacement
Vectors

• ? X0-8 -22-18 -40, ?y0-8 -12-4 -16
• R0-8 43, q 21o N of x axis
• movement in the positive x and y direction,
  change signs to reflect this movement
• ? X8-16 18-2 16 , ?y8-16 4-12 -8
• R8-16 -18, q 26o N of x-component
• movement in the negative x and positive y
  directions, change signs to reflect this movement
• ?x0-16 40-16 24, ?y0-16 168 24
• R0-16 34, q 45o N of x axis

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Warm-Up 09/07/10

1. A plane flies with a velocity of 52 m/s east
   through a 12 m/s cross wind blowing the plane
   south. Find the magnitude and direction (relative
   to due east) of the resultant velocity at which
   it travels.
2. An ambitious hiker walks 25 km west and then 35
   km south in a day. Find the magnitude and
   direction (relative to due west) of her resultant
   displacement.

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• A plane intends to fly north with a speed of 250
  m/s relative to the ground through a high
  altitude cross wind of 50 m/s coming from the
  east. Determine
• The bearing that the plane should take (relative
  to due north and
• The planes speed with respect to the air.