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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