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Chapter 4 Motion in 2 Dimensions

Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

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Page 1: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

Chapter 4

Motion in 2 Dimensions

Page 2: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

Overview

• The focus of this chapter is kinematics in 2-D– Projectile Motion– Uniform Circular Motion– Tangential/Radial Acceleration– Relative Motion

Page 3: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.1 Pos, Vel, Accel Vectors

• Extending what we know about 1-D (straight line) motion to 2-D (motion in xy plane)

• r - position vector (x i + y j )(Points from the origin)

• Δr = rf - ri

displacement vector

(vector subtraction = tail to tail)

Page 4: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.1

• - Average Velocity Vector(points along direction of Δr, t is a scalar)

• - Instantaneous Velocity

– First Derivative of the Position Vector Function with respect to time

t

r

v

dt

d

tt

rrv

0lim

Page 5: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.1

Page 6: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.1

• Acceleration- rate of change of velocity

• - average acceleration

• - instantaneous acceleration

– First Derivative of the Velocity Vector Function– Second Derivative the Position Vector Function

if

if

ttt

vvv

a

dt

d

tt

vva

0lim

Page 7: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.1

• Remember- acceleration = rate of change of vAccel can be cause by changes in

Magnitude (speed)Direction

• Quick Quizzes Pg 80

Page 8: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.2 2-D Motion with cons. Accel

• We can study an object moving in two dimensions if its position vector as a function of time is know.

• - Position function

• - Velocity function

jir ˆ)(ˆ)( tytx

jir

v ˆˆdt

dy

dt

dx

dt

d

jiv ˆˆyx vv

Page 9: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.2

• Example 4.1 Pg 82

Page 10: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.3 Projectile Motion

• Projectile Motion– Easily studied with two assumptions.• Vertical Motion is equivalent to free fall (-g)• Air Resistance is Negligible

– The path of the projectile (trajectory) is parabolic in shape.

– Track the motion as two separate functions• Up and Down (free fall)• Left and Right (uniform motion)

Page 11: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.3

Page 12: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.3

• Quick Quizzes Pg 85• Example 4.2 Pg 85

• Vertical Height (See Board Derivation)• Horizontal Range (see Board Derivation)• Maximum Range (45o)

Page 13: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.3

• Example Problems 4.4-4.7

Page 14: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.4 Uniform Circular Motion

• An object following a circular path at constant speed.

• Acceleration is due to changing direcition of the tangent velocity vector.

Page 15: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.4

• Centripetal Acceleration– Points to the Center of the Circular Path– Perpendicular to the tangent velocity

Quick Quizzes/Example Pg 93

2

C

va

r

T

rv

2 rfv 2

Page 16: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.5 Tangential and Radial Accel

• If the speed of an object is not constant around a circular path– The portion of the acceleration due to changing

direction- radial acceleration– The portion of the acceleration due to changing

speed- tangential acceleration

Page 17: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.5

Page 18: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.5

• Tang. Accel

• Radial Accel

• Total Accel (Magnitude)

• Total Unit Vector Accel

t

da

dt

v

2

r C

va a

r

2 2r ta a a

2ˆ ˆt r

d v

dt r

va a a r

Page 19: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.5

2ˆ ˆt r

d v

dt r

va a a r

Page 20: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.5

• Remember- for uniform circular motionat = 0

Quick Quizzes/Example Pg 95

Page 21: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.6 Relative Velocity and Accel

• How motion is observed from a moving frame of reference rather than fixed frame.

• Airport People Movers (moving sidewalk) Fig 4.21

• Ball and Skateboard Fig 4.22

Page 22: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.6

• The observed displacements and velocities are different to the two observers

• The accelerations however remain the same assuming that the moving frame has constant speed.

Page 23: Chapter 4 Motion in 2 Dimensions. Overview The focus of this chapter is kinematics in 2-D – Projectile Motion – Uniform Circular Motion – Tangential/Radial

4.6

Quick Quiz Pg 98Examples 4.10, 4.11 Pg 98-99