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41
Chapter 3 Motion in Two or Three Dimensions

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Page 1: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

Chapter 3

Motion in Two or Three Dimensions

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Chapter 3

Kinematics in Two or Three Dimensions: position in 3D, velocity in 3D, acceleration in 3D

Projectile Motion

Uniform Circular Motion

Relative motion

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Why do we need to learn projectile motion?

Courtesy: New York Times

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Important Equations of Motion

If the acceleration is constant

Position, velocity and Acceleration are

221

00

0

tatv x x

tav v

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Position in 3 dimensions

kZ(t) jY(t) iX(t) (t)R

kZ jY iX R

dimensions 3in

origin the torelative

position,our as R Write

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Velocity in 3 dimensions

ka ja ia

kdt

dV j

dt

dV i

dtdV

dt

)kV jV id(V

dtVd a

zyx

zyx

zx

y

dtkd

dtjd

0 dt

id used have We :Note

kz

V jy

V ix

V

kdtdZ j

dtdY i

dtdX

dt)kZ jY id(X

dtRd V

then position, theis R If

x

z

yji

k

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Projectile Motion

The horizontal and vertical equations of the motion behave independently

Problem solving:

The trick for all these problems is to break them up into the X and Y directions.

ay=-g and ax=0

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Ball Dropping

ay = g (downwards)

ax = 0

Constant for both cases!!!

vx = 0 vx>0

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X and Y motion are separable—Figure 3.16• The red ball is

dropped, and the yellow ball is fired horizontally as it is dropped.

• The strobe marks equal time intervals.

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Feedback

The most difficult concept was making sure to remember that the y is independent of the x, therefore the time it takes for a projection to hit the ground is dependent on the height it starts at or reaches.

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Equations of motion under constant acceleration

Projectile motion sets xo = 0 and yo = 0 then obtains the specific results shown at right.

x = (vocosαo)t

y = (vosinαo)t 1/2gt2

vx = vocosαo

vy = vosinαo gt

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Firing up in the air at an angleA ball is fired up with velocity Vo and angle θo. What is the range?

y-component

y y0 0

v0 y v0 sin

vy

ay g

t

x-component

x x0 R ?

vx v0 cost

Time of flight can be obtained by the x-component

x x0 R

vx v0 cos

t R

v0 cos

y y0 v0yt 1

2ayt

2

0 v0 sinR

v0 cos

1

2g

R

v0 cos

2

tan 1

2g

R

v02 cos 2

R 2sin cosv0

2

g sin(2)

v02

g

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Question

Which ship gets hit first?

a) 1b) 2c) Both at same time

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Question

Which ship gets hit first?

a) 1b) 2c) Both at same time

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Kick a footballA football is kicked at angle Q0 with respect to the

ground with speed V0. Calculate:

a) The maximum height

b) The velocity at the maximum height

c) The time of travel before the football hits the ground

d) How far away it hits the ground

e) What angle maximizes the distance traveled

Assume the ball leaves the foot at ground level and ignore air resistance

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Rock and Cliff Problem

A ball is thrown horizontally out off a cliff of a height h above the ground. The ball hits the ground a distance Dfrom the base of the cliff.

Assuming the ball was moving horizontally at the top and ignoring air friction, what was its initial velocity?

h

Dy=0

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

An object is dropped from an airplane flying at a constant speed in a straight line at a constant altitude. If there is no air resistance, the falling object will (as seen from the ground)

A. lag behind the airplane, but still move forward.

B. lag behind the airplane and fall straight down.

C. lag behind the airplane and move backward.

D. remain directly under the airplane.

E. move ahead of the airplane.

Q3.13

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Rescue PlaneYou are the pilot of a rescue plane. Your

mission is to drop supplies to isolated mountain climbers on a rocky ridge a height h below. If your plane is traveling horizontally with a speed of VO:

How far in advance of the recipients (horizontal distance) must the goods be dropped?

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Rescue Plane cont.b) Suppose instead, that your plane can’t get to

that position. Instead you must release the supplies a horizontal distance D in advance of the mountain climbers. What vertical velocity should you give the supplies so that they arrive precisely at the the climbers position.

c) What is the speed of the supplies as they hit the ground?

Page 20: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Football PuntA football is kicked at angle Q0 with a velocity V0.

The ball leaves the punters foot h meters above the ground.

How far does it travel, in the X direction, before it hits the ground?

h

Page 21: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

As a ship is approaching the dock at 45.0 cm/s, an important piece of landing equipmentneeds to be thrown to it before it can dock. This equipment is thrown at 15.0 m/s at 60.0o

above the horizontal from the top of a tower at the edge of the water, 8.75 m above theship's deck. For this equipment to land at the front of the ship, at what distance from thedock should the ship be when the equipment is thrown? Air resistance can be neglected.

R

h

y-component

y y0 h

v0 y v0 sin

vy

ay g

t

x-component

x x0 R

vx v0 cost

y y0 v0yt 1

2ayt

2

h v0 sint 12

gt 2

t 2 2v0 sin

gt

2h

g 0

t

2v0 sing

4v 2

0 sin 2

g2 8hg

2 T

T

R v0 cosT

D R vBT

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

10 m

WHAT INITIAL SPEED DOES HE NEED?

First we break up the problem in x and y components

y-component x-component

Time of flight can be obtained by the x-component

Page 23: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Constant Acceleration Review

2y2

10y0

2x2

10x0

tat v y y

tat v x x

ta v v

ta v v

y0yy

x0xx

EACH direction is independent, the only commonality they have is the time

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Components of velocity and accelerationA velocity in the xy-plane can be decomposed into separate x and y components.

Derivative of each component

Page 25: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Uniform Circular MotionFancy words for moving in a circle with constant

speed

We see this around us all the time

• Moon around the earth

• Earth around the sun

• Merry-go-rounds

Page 26: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Uniform Circular Motion - Velocity

Velocity vector = |V| tangent to the circle

Is this ball accelerating?

• Why?

Page 27: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Uniform Circular Motion

A string in a horizontal circle above her head. If the string breaks at the instant shown, which arrow best represents the path the rock will follow?

Page 28: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Centripetal Acceleration

Vector difference V2 - V1 gives the direction of acceleration a

dtvvdtvda /)(/ 12

a

R

Page 29: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Centripetal Acceleration“Center Seeking”

Acceleration vector points towards the center, perpendicular to velocity

Direction changes, mag same

Unlike projectile motion, where a has same mag and direction along path

r a

v 2

R(ˆ r )

R

Page 30: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Acceleration for uniform circular motion

• For uniform circular motion, the instantaneous acceleration always points toward the center of the circle and is called the centripetal acceleration.

• The magnitude of the acceleration is arad = v2/R.

• The period T is the time for one revolution, and arad = 4π2R/T2.

Page 31: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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The acceleration vector: graphical interpretationThe acceleration vector can result in a change in either the magnitude OR the direction of

the velocity.

Page 32: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Figure 3.29

© 2016 Pearson Education, Inc.

Page 33: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Different acceleration scenarios

• Notice the acceleration vector change as velocity decreases, remains the same, or increases as one goes around a curve.

Page 34: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Circular Motion: Get the speed!

Speed = distance/time

Distance in 1 revolution divided by the time it takes to go around once

Speed = 2pr/T

Note: The time to go around once is known as the Period, or T

a = v2/r = (2pr/T)2/r = 4p2r/T2

Page 35: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Ball on a StringA ball at the end of a string is

revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t.

What is the centripetal acceleration?

Page 36: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Relative velocity in one dimension

• If point P is moving relative to reference frame A, we denote the velocity of P relative to frame Aas vP/A.

• If P is moving relative to frame B and frame B is moving relative to frame A, then the x-velocity of P relative to frame A is vP/A-x

= vP/B-x + vB/A-x.

Page 37: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Page 38: 0RWLRQ LQ 7ZR RU 7KUHH 'LPHQVLRQVpeople.tamu.edu/~mahapatra/teaching/ch3.pdfZ ] ] } Z ] v P Z } l ð ñ X ì u l U v ] u } v ] } ( o v ] v P µ ] u v v } Z } Á v } ] ( } ] v } l Xd

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Relative velocity in two or three dimensions

• We extend relative velocity to two or three dimensions by using vector addition to combine velocities.

• In Figure 3.34, a passenger’s motion is viewed in the frame of the train and the cyclist.

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Flying in a crosswind• A crosswind affects the motion of an airplane.

• An airplane’s compass indicates it is headed north, and its airspeed is 240 km/hr. If there is a 100 km/hr wind from west to east, what is the velocity of the airplane relative to earth?

In which direction should the pilot head to travel due north and what will be her velocity relative to earth?

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Relative Velocity: Boat on the RiverYou want to cross the

river so that the boat gets exactly from A to B. The river has a current vC=4 km/h. Your boat’s speed in still water is vB=20km/h?

What is the angle you should aim at to do that?

How long would it take?

vB

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Heading North

You are the navigator of a flight scheduled to fly from New Orleans due north to St. Louis, a distance of 673miles. Your instruments tell you that there is a steady wind from the northwest with a speed of 105 mph. The pilot sets the air speed at 575 mph and asks you to find the estimated flying time. What do you tell her?


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