Two or Three Dimensional Motion

1
Chapter 3

• Two or Three Dimensional Motion

2
Position, Velocity and Acceleration Vectors

• Coordinates in the x,y,z plane
• (x,y,z)
• Unit Vectors
• Position Vector

3
Average and Instantaneous Velocity
Components
4
Components, Magnitude and Direction
Example 3.1
5
Acceleration Vector

• Circular Motion Example

6
Components

• Unit Vectors
• Second Derivatives
• Example 3.2

7
Parallel and Perpendicular Components

• is in the same direction as the velocity vector.
• means a change in the velocity's magnitude.
• is 90o from the velocity vector.
• means a change in the velocity's direction.

8
Parallel and Perpendicular Components

• What acceleration component is responsible for
• Superman slowing down a speeding train?
• Mark Martin accelerating while passing Ricky
  Martin along a straight stretch of I-70?
• Superman spinning Ricky Martin in a circular path
  at a constant speed.
• Example 3.4

9
Projectile Motion

• Projectile any body that is given an initial
  velocity and then follows a trajectory determined
  entirely by gravity and air resistance.
• Air Resistance and the curvature of the earth
  will be ignored unless otherwise stated.

10
Kinematic Equations for Free Fall, x and y
flavors

• y

• x

11
Projectile Motion Equations
Insert the component equations into the x and y
flavors of the kinematic equations
12
Magnitude and Direction

• Pythagorean
• Position
• Velocity
• Direction of Velocity
• Examples 3.7 and 3.8

13
Motion in a Circle

• Uniform Circular Motion when a particle moves in
  a circle with a constant speed
• Follow derivation of radial acceleration on pg 98.

Example 3.12
14
Non-Uniform Circular Motion

• Circular motion where velocity is not constant.
• arad is still perpendicular to the motion and
  still found by
• New parallel component to the acceleration, atan

15
Relative Velocity

• The velocity seen by a particular observer
• What is the velocity of a woman walking 1.00 m/s
  along a train car, as seen by an observer on the
  ground
• If the observer is stationary (v 0.00 m/s)
• If the observer is moving
• V 1.00 m/s
• V -1.00 m/s

16
Velocity depends on the observers and objects
velocities

• vP/A is the velocity of the object (P) in the
  observers reference frame (A)
• vP/B is the velocity of the object (P) in the
  objects reference frame (B)
• vB/A is the velocity of the objects frame (B) as
  seen by the observer (A)

17
Relative Motion in 2 or 3D

• Instead of x coordinates, use position vectors (
  ).
• Taking the time derivative yields
• Pythagorean Theorem gives magnitude and direction
  is found by
• Example 3.15