Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative...
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Transcript of Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative...
![Page 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.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/1.jpg)
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
![Page 2: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/2.jpg)
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
![Page 3: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/3.jpg)
Position and Velocity
1 1x i y j
r xi yj
dr dx dyV i j
dt dt dt
![Page 4: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/4.jpg)
Instantaneous velocityThe Instantaneous velocity vectoris tangent to the trajectory.The direction of the velocity is to the curve.
![Page 5: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/5.jpg)
Don’t confuse these two graphs
sds
Vdt
2 2( ) ( )dx dy
Vdt dt
![Page 6: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/6.jpg)
Acceleration
avgV
at
dVa
dt
![Page 7: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/7.jpg)
The instantaneous acceleration can bedecomposed into parallel and perpendicular components
![Page 8: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/8.jpg)
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
![Page 9: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/9.jpg)
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?
![Page 10: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/10.jpg)
A projectile launched horizontally falls in the same time as projectile that is released from rest
![Page 11: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/11.jpg)
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
![Page 12: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/12.jpg)
Launch angle
cos
sin
ix i
iy i
V V
V V
21/ 2 ( )
ix
iy
x V t
y V t g t
![Page 13: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/13.jpg)
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
![Page 14: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/14.jpg)
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
![Page 15: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/15.jpg)
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
![Page 16: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/16.jpg)
Trajectories of a projectile launched at different angles with the same speed
![Page 17: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/17.jpg)
Relative Motion
Relative position
Relative velocity
'r r R
ab ac cbV V V
![Page 18: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/18.jpg)
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
![Page 19: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/19.jpg)
Angular Velocity
Average angular velocity =∆θ/∆t
Instantaneous angular velocity
The angular velocity is constant during uniform circular motion
d
dt
t 2
T
![Page 20: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/20.jpg)
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
![Page 21: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/21.jpg)
Velocity and acceleration in uniform circular motion
![Page 22: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/22.jpg)
Velocity in uniform circular motion
Has only a tangentialComponent
The magnitude of velocity is a constant
Vt =r dθ/dt =ωr
![Page 23: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/23.jpg)
Centripetal acceleration
![Page 24: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/24.jpg)
The magnitude of centripetal acceleration
22
rV
a rr
P184
Towards center of circle
![Page 25: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/25.jpg)
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
![Page 26: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/26.jpg)
Nonuniform Circular Motion
tdV
adt
Change the speed
d with =
dtV r
ta rHere α is angular acceleration
if is constantf i t
![Page 27: Chapter 4:Kinematics in Two Dimensions 1.Two-Dimension Kinematics 2.Projectile Motion 3.Relative Motion 4.Uniform Circular Motion 5.Velocity and Acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022081421/56649ee55503460f94bf4d5f/html5/thumbnails/27.jpg)
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