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

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

Position and Velocity

1 1x i y j

r xi yj

dr dx dyV i j

dt dt dt

Instantaneous velocityThe Instantaneous velocity vectoris tangent to the trajectory.The direction of the velocity is to the curve.

Don’t confuse these two graphs

sds

Vdt

2 2( ) ( )dx dy

Vdt dt

Acceleration

avgV

at

dVa

dt

The instantaneous acceleration can bedecomposed into parallel and perpendicular components

Stop to think:This acceleration will cause the particle to:

a. Speed up and curve upwardb. Speed up and curve downwardc. Slow down and curve upwardd. Slow down and curve downwarde. Move to the right and downf. Reverse direction

Projectile Motionobject moves in two dimensions under the gravitational force.

0x

y

a

a g

A

B

1. What is the accelerations at position A and B?2. What is the velocities at position A and B?

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

Plot of projectile motion in t-xy

01

2

3

4

0

2

4

6

8

10

12

14

16

18

20

020

4060

80100

120140

Launch angle

cos

sin

ix i

iy i

V V

V V

21/ 2 ( )

ix

iy

x V t

y V t g t

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

Problem 50

6sin( 15 ) /ooyV m s

6cos( 15 ) /ooxV m s

23 1/ 2oyy V t gt

Solve a quadratic equation to get t

*oxd V t

24.9 1.55 3 0t t

The maximum height and distance of fly ball For projectile motion, always

remember:

g

vh ii

2

sin 22

0, x ya a g

g

vR ii 2sin2

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

Relative Motion

Relative position

Relative velocity

'r r R

ab ac cbV V V

Uniform Circular Motion

Period

Angular Position

1 circumference

speedT

2 rT

V

(radians)s

r

full circle2 r

= 2 radr

3601 rad 57.3

2

oo

Angular Velocity

Average angular velocity =∆θ/∆t

Instantaneous angular velocity

The angular velocity is constant during uniform circular motion

d

dt

t 2

T

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

2

T

2T

Velocity and acceleration in uniform circular motion

Velocity in uniform circular motion

Has only a tangentialComponent

The magnitude of velocity is a constant

Vt =r dθ/dt =ωr

Centripetal acceleration

The magnitude of centripetal acceleration

22

rV

a rr

P184

Towards center of circle

Velocity and acceleration in Uniform Circular Motion

The velocity has only a tangential component Vt

(with in rad/s)tds d

V r rdt dt

2

(toward center of ciecle)V

ar

Nonuniform Circular Motion

tdV

adt

Change the speed

d with =

dtV r

ta rHere α is angular acceleration

if is constantf i t

Rotational kinematics

For constant angular acceleration

2

i ff i t

f i t 21/ 2 ( )f i i t t

2 2 2f i