Chapter 4:Kinematics in Two Dimensions

1
Chapter 4Kinematics 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 93Stop to think 4.2 P
97Stop to think 4.3 P 102Stop to think 4.4 P
107Stop to think 4.5 P 110Stop to think 4.6
P 113

• Example 4.3 P97
• Example 4.4 P98
• Example 4.5 P100
• Example 4.6 P101
• Example 4.9 P106
• Example 4.13 P110
• Example 4.15 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
Dont confuse these two graphs
6
Acceleration
7
The instantaneous acceleration can bedecomposed
into parallel and perpendicular components
8
Stop to thinkThis acceleration will cause the
particle to

1. Speed up and curve upward
2. Speed up and curve downward
3. Slow down and curve upward
4. Slow down and curve downward
5. Move to the right and down
6. Reverse direction

9
Projectile Motionobject moves in two dimensions
under the gravitational force.
B
A

1. What is the accelerations at position A and B?
2. 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 VxThere is no acceleration at x
direction.ie. R Vxt, t R/Vx(2) Viy0, 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 ??/?t
• Instantaneous angular velocity
• The angular velocity is constant during uniform
  circular motion

20
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 2p/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
d?/dt ?r
Has only a tangential Component
23
Centripetal acceleration
24
The magnitude of centripetal acceleration
P184
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 a is angular acceleration
27
Rotational kinematics

• For constant angular acceleration