Motion in Two Dimensions Ch 4 Sect' 41

1
Two-Dimensional Kinematics(Walker Ch. 4)

• Motion in Two Dimensions (Ch 4 Sect. 4-1)
• Projectile Motion (Ch 4 Sect. 4-2)
• Zero Launch Angle (Ch 4 Sect. 4-3)

2
Motion in Two Dimensions

• Here we study the kinematics of an object that
  can be modeled as a particle moving in a plane.
• Occurs in many practical situations
• Projectile motion
• Satellites
• Electron orbits
• The main idea of the chapter
• Motion can be divided into 2 components
  horizontal and vertical - one motion in the
  x-direction and the other in the y-direction
• Horizontal and vertical motions are independent

3
Producing An Acceleration
v 0 for an instant

• Various changes in a particles motion may
  produce an acceleration
• The magnitude of the velocity vector may change
• The direction of the velocity vector may change
  (even if the magnitude remains constant)
• Both may change simultaneously

4
Two-Dimensional Motion with Constant Acceleration

• When the two-dimensional motion has a constant
  acceleration, a series of equations can be
  developed that describe the motion
• These equations will be similar to those of
  one-dimensional kinematics

5
Motion in Two Dimensions - Kinematic Equations

• Position vector
• Velocity
• Since acceleration is constant, we can also find
  an expression for the velocity as a function of
  time

6
Kinematic Equations, Component Equations

• becomes
• vxf vxi axt and
• vyf vyi ayt
• 
  becomes
• xf xi vxi t 1/2 axt2 and
• yf yi vyi t 1/2 ayt2

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Constant-Acceleration Equations of
MotionSummaryTable 4-1 from Walker
These equations will be used for all of the
results in Ch. 4!
8
Two-Dimensional Motion Example

• Problem 3 p.98. Starting from rest, a car
  accelerates at up a hill that is inclined 5.5
  above the horizontal. How far (a) horizontally
  and (b) vertically has the car traveled in 12 s?
• Problem 4 p.98. A particle passes through the
  origin with a velocity of If
  the particles acceleration is
• (a) what are its x and y positions after 5.0 s?
• (b) What are vx and vy at this time?
• (c) Does the speed of this particle increase with
  time, decrease with time, or increase and then
  decrease? Explain.

9
Projectile Motion (Ch. 4 Sect. 4-2)

• An object may move in both the x and y directions
  simultaneously.
• The form of two-dimensional motion we will deal
  with is called projectile motion.
• A projectile is an object that is thrown,
  kicked, batted, or otherwise launched into motion
  and then allowed to follow a path determined
  solely by the influence of gravity (from Walker
  p.81)

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Assumptions of Projectile Motion

• The free-fall acceleration is constant over
  the range of motion
• And is directed downward
• The effect of air friction is negligible
• With these assumptions, an object in projectile
  motion will follow a parabolic path
• This path is called the trajectory

11
Projectile Motion

• No acceleration in x-direction! (ax0)

12
Projectile Motion Equations
- g
13
Projectile Motion Conceptual Question

• What is the acceleration of a projectile when it
  reaches its highest point? What is its
  acceleration just before and just after reaching
  this point?

14
Zero Launch Angle (Ch 4 Sect. 4-3)
x0 0, y0 h v0x v0 cos ? v0y v0 sin ?
x0 0, y0 h v0x v0 v0y 0

• The x-component of velocity remains the same!
• The y-component of velocity changes in the same
  way as at a free fall motion

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Zero Launch Angle
x0 0, y0 h v0x v0, v0y 0
16
Zero Launch Angle
17
Zero Launch Angle Example

• Problem 13 p. 99. A crow is flying horizontally
  with a constant speed of 2.70 m/s when it
  releases a clam from its beak. The clam lands on
  the rocky beach 2.10 s later. Just before the
  clam lands, what is
• (a) its horizontal component of velocity, and
• (b) its vertical component of velocity?
• (c) How would your answers to parts (a) and (b)
  change if the speed of the crow were increased?
  Explain

18

• A car and a truck are looking for a parking spot
  in a parking lot. The car is moving with a speed
  of 8.50 m/s due North relative to the ground
  while the truck is moving with a speed of 6.35
  m/s in a direction 20o North of West relative to
  the ground.
• (a) Sketch the velocity vector for each vehicle,
  and write it in unit vector notation.
• (b) What is the velocity of the car relative to
  the truck?
• (c) What is the velocity of the truck relative to
  the car?