1. Chapter 4
Motion in 2 Dimensions

2. Overview The focus of this chapter is kinematics in 2-D
Projectile Motion
Uniform Circular Motion
Tangential/Radial Acceleration
Relative Motion

3. 4.1 Pos, Vel, Accel Vectors
Extending what we know about 1-D (straight line) motion to 2-D (motion in xy plane)
r - position vector (x i + y j )
(Points from the origin)
Δr = rf - ri
displacement vector
(vector subtraction = tail to tail)

4. 4.1 - Average Velocity Vector - Instantaneous Velocity
(points along direction of Δr, t is a scalar)
- Instantaneous Velocity
First Derivative of the Position Vector Function with respect to time

5. 4.1

6. 4.1 Acceleration- rate of change of velocity - average acceleration
- instantaneous acceleration
First Derivative of the Velocity Vector Function
Second Derivative the Position Vector Function

7. 4.1 Remember- acceleration = rate of change of v Quick Quizzes Pg 80
Accel can be cause by changes in
Magnitude (speed)
Direction
Quick Quizzes Pg 80

8. 4.2 2-D Motion with cons. Accel
We can study an object moving in two dimensions if its position vector as a function of time is know.
- Position function
- Velocity function

9. 4.2
Example 4.1 Pg 82

10. 4.3 Projectile Motion Projectile Motion
Easily studied with two assumptions.
Vertical Motion is equivalent to free fall (-g)
Air Resistance is Negligible
The path of the projectile (trajectory) is parabolic in shape.
Track the motion as two separate functions
Up and Down (free fall)
Left and Right (uniform motion)

11. 4.3

12. 4.3 Quick Quizzes Pg 85 Example 4.2 Pg 85
Vertical Height (See Board Derivation)
Horizontal Range (see Board Derivation)
Maximum Range (45o)

13. 4.3
Example Problems

14. 4.4 Uniform Circular Motion
An object following a circular path at constant speed.
Acceleration is due to changing direcition of the tangent velocity vector.

15. 4.4 Centripetal Acceleration Quick Quizzes/Example Pg 93
Points to the Center of the Circular Path
Perpendicular to the tangent velocity
Quick Quizzes/Example Pg 93

16. 4.5 Tangential and Radial Accel
If the speed of an object is not constant around a circular path
The portion of the acceleration due to changing direction- radial acceleration
The portion of the acceleration due to changing speed- tangential acceleration

17. 4.5

18. 4.5 Tang. Accel Radial Accel Total Accel (Magnitude)
Total Unit Vector Accel

19. 4.5

20. 4.5 Remember- for uniform circular motion at = 0
Quick Quizzes/Example Pg 95

21. 4.6 Relative Velocity and Accel
How motion is observed from a moving frame of reference rather than fixed frame.
Airport People Movers (moving sidewalk) Fig 4.21
Ball and Skateboard Fig 4.22

22. 4.6
The observed displacements and velocities are different to the two observers
The accelerations however remain the same assuming that the moving frame has constant speed.

23. 4.6
Quick Quiz Pg 98 Examples 4.10, 4.11 Pg 98-99