Honors Physics Chapter 3

1
Honors Physics Chapter 3

• Chapter 3 2-D Motion
• Vectors
• Vector Operations
• Projectile Motion
• Circular Motion

2
Vectors

• Vectors are graphically represented by arrows

• The direction of the physical quantity is given
  by
• the direction of the arrow.
• The magnitude of the quantity is given by the
• length of the arrow.

3
Addition of Vectors

• Graphical Tail-to-head method
• Resultant of Velocities (Addition of Vectors)

4
Graphical Method - Example

• You are told to walk due east for 50 paces, then
• north for 38 paces, and then due south
• for 30 paces.
• What is the magnitude and direction of your total
• displacement ?

5
Addition of Vectors

• Using components (A,B lie in x,y plane)
• C AB Ax Ay Bx By CxCy
• Cx and Cy are called vector components of C.
• They are two perpendicular vectors that are
  parallel
• to the x and y axis.
• Ax,Ay and Bx, By are vector components of A and
  B.

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Scalar Components of a Vector (in 2 dim.)

• Vector components of vector A
• A Ax Ay
• Scalar components of vector A
• A Ax x Ay y
• Ax and Ay are called scalar
• components of A.
• x and y are unit vectors.
• Equivalently
• A(Ax,Ay)
• A is a vector pointing from the
• origin to the point with
• coordinates Ax,Ay.

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Scalar Components of a Vector (in 2 dim.)

• Scalar components of vector A
• A Ax x Ay y
• A, q known
• Ax A Cos q
• AyA Sin q
• Ax, Ay known
• A2(Ax )2(AY)2
• q Tan-1 Ay/Ax

8
Addition of Vectors

• Using scalar components (A,B lie in x,y plane)
• C AB Ax x Ay y Bx x By y
  Cx xCy y
• 1. Determine scalar components of A and B.
• 2. Calculate scalar components of C
• Cx AxBx and CyAyBy
• 3. Calculate C and q
• C2(Cx )2(CY)2 q Tan-1 Cy/Cx

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Addition of Vectors

• Vector sum or Resultant

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Displacement and Distance

• Displacement is the vector that points from a
  bodys initial position to its final position.
  The length of is equal to the shortest distance
  between the two positions.
• ?x x x0
• The length of ?x is not the same as distance
  traveled !

11
Average Speed and Velocity

• Average velocity describes how the displacement
  of an object changes over time
• average velocity displacement/elapsed time
• v (x-x0) / (t-t0) ?x / ?t
• Average velocity also takes into account the
  direction of
• motion.
• The magnitude of v is not the same as the
  average speed !

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Review of Concepts Chapter 2

• kinematics A description of motion
• position your coordinates
• displacement ?x change of position (vector)
• velocity rate of change of position (vector)
• average ?x/?t
• instantaneous slope of x vs. t
• acceleration rate of change of velocity
  (vector)
• average ?v/?t

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Free Fall - Symmetry

• At a given displacement along the path of motion
  the
• magnitude of the upward velocity is equal the
  
• magnitude of the downward velocity and they
  point in
• opposite directions
• vup - vdown

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Kinematics in Two DimensionsConstant Acceleration

• Consider an object which moves in the (x,y) plane
  from the initial
• position r0, at time t0 with velocity v0, with
  constant acceleration.
• position your coordinates (just r(x,y) in 2-D)
• displacement ?r r-r0 change of position
• velocity rate of change of position
• average ?r/?t
• instantaneous lim ?t-gt0 ?r/?t
• acceleration rate of change of velocity
• average ?v/?t
• instantaneous lim ?t-gt0 ?v/?t
• Same concepts as in one dimension !
• Equations of kinematics are derived for the x and
  y components
• separately. Same equations as in one dimension !

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Equations of Kinematics in 2 Dim.
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Eqs. of Kinematics in 2 Dim.

• The motions along the x and y directions are
  completely
• independent. They only share a common time.
• Three swimmers can swim equally fast relative to
  the water. They have a race to see who can swim
  across a river in the least time. Relative to
  the water, Beth (B) swims perpendicular to the
  flow, Ann (A) swims upstream, and Carly (C) swims
  downstream. Which swimmer wins the race?
• A) Ann
• B) Beth
• C) Carly
• Time to get across width of river /
  y-component of velocity

correct
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Projectile Motion

• A flatbed railroad car is moving along a track at
  constant
• velocity. A passenger at the center of the car
  throws a ball
• straight up. Neglecting air resistance, where
  will the ball land ?
• 1. Forward of the center of the car
• 2. At the center of the car
• 3. Backward of the center of the car

correct
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Kinematics of Projectile Motion (t00)

• x direction motion with constant velocity gt
  ax 0
• x x0 v0xt
• vx v0x
• y direction free fall gt ay - g -9.80 m/s
• y y0 v0y t - 1/2 g t2
• vy v0y g t
• vy2 v0y2 2 g (y-y0)

19
Circular Motion
Centripetal acceleration
Acceleration is the result of a change in the
velocitys direction. In case of ac this change
is toward the center of the circular motion.
New Definitions frequency f, period T
The period T is the time required to travel once
around the circle, (i.e. to make one complete
revolution T 2 p R/v ) The Frequency f 1
/ T