1. Chapter 4:Kinematics in Two Dimensions
1.Two-Dimension Kinematics
2.Projectile Motion
3.Relative Motion
4.Uniform Circular Motion
5.Velocity and Acceleration in Uniform Circular Motion
6.Nonuniform Circular Motion

2. Stop to think 4. 1. P 93 Stop to think 4. 2. P 97 Stop to think 4. 3
Stop to think 4.1 P 93 Stop to think 4.2 P 97 Stop to think 4.3 P 102 Stop to think 4.4 P 107 Stop to think P 110 Stop to think P 113
Example 4.3 P97
Example 4.4 P98
Example 4.5 P100
Example P101
Example P106
Example P110
Example P114

3. Position and Velocity

4. Instantaneous velocity
The Instantaneous velocity vector
is tangent to the trajectory.
The direction of the velocity is
to the curve.

5. Don’t confuse these two graphs

6. Acceleration

7. The instantaneous acceleration can be decomposed into parallel and perpendicular components

8. Stop to think: This acceleration will cause the particle to:
Speed up and curve upward
Speed up and curve downward
Slow down and curve upward
Slow down and curve downward
Move to the right and down
Reverse direction

9. Projectile Motion object moves in two dimensions under the gravitational force.
What is the accelerations at position A and B?
What is the velocities at position A and B?

10. A projectile launched horizontally falls in the same time as projectile that is released from rest

11. Plot of projectile motion in t-xy

12. Launch angle

13. Ex. A ball thrown horizontally at velocity Vi , travels a horizontal distance of R m before hitting the ground. From what height was the ball thrown? (1) Since ball is thrown horizontally, Vi =Vx There is no acceleration at x direction. ie. R = Vxt, t = R/Vx (2) Viy=0, h = -1/2gt2

14. Problem 50
Solve a quadratic equation to get t

15. The maximum height and distance of fly ball
For projectile motion, always remember:

16. Trajectories of a projectile launched at different angles with the same speed

17. Relative Motion
Relative position
Relative velocity

18. Uniform Circular Motion
Period
Angular Position

19. Angular Velocity Average angular velocity =∆θ/∆t
Instantaneous angular velocity
The angular velocity is constant during uniform circular motion

20. rpm: revolution per minute. 1 rpm = 2π/60 (rad)/s
An old-fashioned single-play vinyl record rotates 30.0 rpm . What are (a) the angular velocity in rad/s and (b) the period of the motion?
rpm: revolution per minute.
1 rpm = 2π/60 (rad)/s

21. Velocity and acceleration in uniform circular motion

22. Velocity in uniform circular motion
The magnitude of velocity is a constant
Vt =r dθ/dt =ωr
Has only a tangential
Component

23. Centripetal acceleration

24. The magnitude of centripetal acceleration
Towards center of circle

25. Velocity and acceleration in Uniform Circular Motion
The velocity has only a tangential component Vt

26. Nonuniform Circular Motion
Change the speed
Here α is angular acceleration

27. Rotational kinematics
For constant angular acceleration