ch3twodimensionalkinematicsonline-101013171228-phpapp02

7/27/2019 ch3twodimensionalkinematicsonline-101013171228-phpapp02

1/68

Newtonian Mechanics

Chapter 3

Kinematics in Two Dimensions


2/68

Learning Objectives

Newtonian Mechanics-KinematicsI.A.2. Motion in two dimensions, including projectile motion

a) Students should be able to add, subtract, and resolve displacementand velocity vectors, so they can:

1) Determine components of a vector along two specified, mutuallyperpendicular axes.

2) Determine the net displacement of a particle or the location of aparticle relative to another.

3) Determine the change in velocity of a particle or the velocity ofone particle relative to another.

b) Students should understand the motion of projectiles in a uniform

gravitational field, so they can:1) Write down expressions for the horizontal and verticalcomponents of velocity and position as functions of time, andsketch or identify graphs of these components.

2) Use these expressions in analyzing the motion of a projectile thatis projected with an arbitrary initial velocity.


3/68

Table of Contents

3.1 Displacement, Velocity and Acceleration

3.2 Equations of Kinematics in Two Dimensions

3.3 Projectile Motion

3.4 Relative Velocity


4/68

Chapter 3Motion in Two Dimensions

Section 1:

Displacement, Veloc ity, and

Accelerat ion


5/68

Motion in Two Dimensions

How do we describe an object moving across a

surface (x & y directions)?

Similar to how we described motion along a line.

Define a reference point, axes of measurement,

and positive directions.

Each Dimension is independent of the other

Think Etch-a-Sketch


6/68

Ahhh the memories


7/68

positioninitialor

positionfinalr

ntdisplaceme orrr

Displacement


8/68

ttt o

o

rrrv

Average veloc i ty is the

displacement divided by

the elapsed time.

Velocity


9/68

The instantaneous

veloci ty indicates how fast

the car moves and the

direction of motion at each

instant of time.

tt

rv

0lim

Velocity


10/68

ttt o

o

vvva

DEFINITION OF AVERAGE ACCELERATION

ov

v v

Acceleration


11/68

Question #1A truck drives due south for 1.2 km in 1.5 minutes. Then, thetruck turns and drives 1.2 km due west for 1.5 minutes. Whichone of the following statements is correct?

a) The average speed for the two segments is the same. Theaverage velocity for the two segments is the same.

b) The average speed for the two segments is not the same.

The average velocity for the two segments is the same.

c) The average speed for the two segments is the same. Theaverage velocity for the two segments is not the same.

d) The average speed for the two segments is not the same.

The average velocity for the two segments is not the same.


12/68

Question #2

A ball is rolling down one hill and up another as shown. Points A and Bare at the same height. How do the velocity and acceleration change asthe ball rolls from point A to point B?

a) The velocity and acceleration are the same at both points.

b) The velocity and the magnitude of the acceleration are the same at bothpoints, but the direction of the acceleration is opposite at B to the directionit had at A.

c) The acceleration and the magnitude of the velocity are the same at bothpoints, but the direction of the velocity is opposite at B to the direction ithad at A.

d) The horizontal component of the velocity is the same at points A and B,

but the vertical component of the velocity has the same magnitude, butthe opposite sign at B. The acceleration at points A and B is the same.

e) The vertical component of the velocity is the same at points A and B, butthe horizontal component of the velocity has the same magnitude, but theopposite sign at B. The acceleration at points A and B has the samemagnitude, but opposite direction.


13/68


Section 2:

Equat ions of Kinemat ics in Two

Dimensions


14/68

Equations of Kinematics

2

2

1 attvxxoo

atvv o

oo xxavv 222

Quick Review


15/68

Keys to Solving 2-D Problems

1. Resolve all vectors into components

x-component

Y-component

2. Work the problem as two one-dimensionalproblems.

Each dimension can obey differentequations of motion.

3. Re-combine the results for the two componentsat the end of the problem.


16/68

tavv xoxx oxoxx xxavv 2

22

221

, tatvxx xxoo

x component


17/68

tavv yoyy

2

21 tatvyy yoyo

oyoyy yyavv 222

y component


18/68

Actual motion

The x part of the motion occurs exactly as it would if the y

part did not occur at all, and vice versa.

Remember the bullet drop video!


19/68

In thexdirection, the spacecraft has an initial velocity

component of +22 m/s and an acceleration of +24 m/s2. In the

ydirection, the analogous quantities are +14 m/s and an

acceleration of +12 m/s2

. Find (a) xand vx, (b) yand vy,and (c) the final velocity of the spacecraft at time 7.0 s.

Example 1 A MovingSpacecraft


20/68

In thexdirection, the spacecraft has an initial velocity

component of +22 m/s and an acceleration of +24 m/s2. In the

ydirection, the analogous quantities are +14 m/s and an

acceleration of +12 m/s2

. Find (a)

xand vx, (b)

yand vy,and (c) the final velocity of the spacecraft at time 7.0 s.

x ax vx vox t

? +24.0 m/s2 ? +22 m/s 7.0 s

y ay vy voy t

? +12.0 m/s2 ? +14 m/s 7.0 s

Example 1 A Moving Spacecraft


21/68

x ax vx vox t

? +24.0 m/s2 ? +22 m/s 7.0 s

m740s0.7sm24s0.7sm22 22

2

1

2

21

tatvx xox

sm190s0.7sm24sm22 2 tavv xoxx

x component


22/68

y ay vy voy t

? +12.0 m/s2 ? +14 m/s 7.0 s

m390s0.7sm12s0.7sm14 22

2

1

2

21

tatvy yoy

sm98s0.7sm12sm14 2 tavv yoyy

y component


23/68

v

sm98yv

sm190x

v

sm210sm98sm190 22 v

2719098tan 1

Resultant velocity


24/68

Question #3

An eagle takes off from a tree branch on the side of amountain and flies due west for 225 m in 19 s. Spying amouse on the ground to the west, the eagle dives 441 m at anangle of 65relative to the horizontal direction for 11 s to catch

the mouse. Determine the eagles average velocity for thethirty second interval.

a) 19 m/s at 44below the horizontal direction

b) 22 m/s at 65below the horizontal direction

c) 19 m/s at 65below the horizontal direction

d) 22 m/s at 44below the horizontal direction

e) 25 m/s at 27below the horizontal direction


25/68

Question #4

In two-dimensional motion in thex-yplane, what is therelationship between thexpart of the motion to theypart ofthe motion?

a) Thexpart of the motion is independent of the ypart of the

motion.b) The ypart of the motion goes as the square of thexpart of

the motion.

c) Thexpart of the motion is linearly dependent on the ypart ofthe motion.

d) Thexpart of the motion goes as the square of the ypart ofthe motion.

e) If the ypart of the motion is in the vertical direction, thenxpartof the motion is dependent on the ypart.


26/68

Question #5

Complete the following statement: In two-dimensionalmotion in thex-yplane, thexpart of the motion and the ypart of the motion are independent

a) only if there is no acceleration in either direction.

b) only if there is no acceleration in one of the directions.

c) only if there is an acceleration in both directions.

d) whether or not there is an acceleration in any direction.

e) whenever the acceleration is in the ydirection only.


27/68

Question #6

A roller coaster car rolls from

rest down a 20oincline with an

acceleration of 5.0 m/s2

How far horizontally has thecoaster travelled in 10

seconds?

a = 5.0 m/s2

a

ax

ay


28/68

Question #7

A roller coaster car rolls from

rest down a 20oincline

with an acceleration of

5.0 m/s2

How far vertically has the

coaster travelled in 10

seconds?

a = 5.0 m/s2

a

ax

ay


29/68

Question #8

Jackson heads east at 25 km/h for 20 minutes beforeheading south at 45 km/h for 20 minutes. Hunter headssouth at 45 km/h for 10 minutes before heading east at25 km/h for 30 minutes. Which driver has the greater

average velocity, if either?

a) Jackson

b) Hunter

c) They both have the same average velocity.


30/68


Section 3:

Project i le Motion


31/68

Projectile Motion

Something is fired, thrown, shot, or hurled near the

Earths surface

Horizontal velocity is constant

ax= 0, vo,x= vx Vertical velocity is accelerated

ay= g = 9.80665 m/s2 10 m/s2(constant) Air resistance is ignored

If launch angle is not horizontal or vertical

must resolve initial velocity into x and y components


32/68

The airplane is moving horizontally with a constant velocity of

+115 m/s at an altitude of 1050 m. Determine the time required

for the care package to hit the ground.

Example 3 A Falling Care Package


33/68

y ay vy voy t

-1050 m -9.80 m/s2 0 m/s ?


34/68

y ay vy voy t

-1050 m -9.80 m/s

2

0 m/s ?

2

2

1 tatvyyoy

22

1 tay

y

yayt

2

0


35/68

Example 4The Velocity of the Care Package

A plane is moving horizontally with a constant

velocity of 115 m/s at an altitude of 1050 m.

Ignoring air resistance, determine the velocity of

a package just before it hits the ground.


36/68

yavv yyoy 22,

2

sm184v

22

yx vvv

yavvv yoxo 22

,

2

,

xox vv ,

msmsmsmv 105081.920115 222

v

vxcos

vvx1cos

o3.51

y ay v vo,y vo,x t

-1050 m -9.80 m/s2

? 0 m/s 115 m/s 14.6 s

smsm

184115


37/68

Conceptual Example 5

I Shot a Bullet into the Air

Suppose you are driving a

convertible with the top

down. The car is moving to

the right at constant velocity.You point a rifle straight up

into the air and fire it. In the

absence of air resistance,

where would the bullet land

(a) behind you(b) ahead of you

(c) in the barrel of the rifle


38/68

A placekicker kicks a football at and angle of 40.0 degrees and

the initial speed of the ball is 22 m/s. Ignoring air resistance,

determine the maximum height that the ball attains.

Example 6 The Height of aKickoff


39/68

ov

oxv

oyv

sm1440sinsm22sin ooy vv

sm1740cossm22sin oox vv


40/68

y ay vy voy t

? -9.80 m/s2 0 14 m/s


41/68

3 3 Projectile Motion

yavv yyoy

2

2

,

2

y ay vy voy t

? -9.80 m/s2 0 14 m/s

y

oyy

a

vv

y 2

22

2

2

sm8.92sm140

y m10


42/68

What is the time of flight between kickoff and landing?

Example 7The Time of Flight of a Kickoff


43/68

y ay vy voy t

0 -9.80 m/s2 14 m/s ?

2

21

, tatvy yyo

s9.2t

tavty yyo 21

,0

s0ttav yyo 2

1,0

y

yo

a

vt ,

2

280.9142 smsm


44/68

Calculate the range R of the projectile.

Example 8The Range of a Kickoff


45/68

x ax vx Vo,x t

? 0 17 m/s 2.9 s

2

21

, tatv xxox

mx 49

ssmx 9.217

0


46/68

Question #9

A diver running 1.8 m/s dives out horizontally from

the edge of a vertical cliff and 3.0 s later reaches the

water below. How high was the cliff?

h

d

v


47/68

Question #10

A diver running 1.8 m/s dives out horizontally from the

edge of a vertical cliff and 3.0 s later reaches the water

below. How far from its base did the diver hit the water?

h

d

v


48/68

Question #11

When a football in a field goal attempt reaches

its maximum height, how does its speed

compare to its initial speed?

a) It is zero.

b) It is equal to its initial speed.

c) It is greater than its initial speed.d) It is less than its initial speed.


49/68

Question #12

The Zambezi River flows over Victoria Falls in Africa. The

falls are approximately 108 m high. If the river is flowing

horizontally at 3.6 m/s just before going over the falls,

what is the speed of the water when it hits the bottom?

Assume the water is in freefall as it drops.


50/68

Question #13

An astronaut on the planet Zircon tosses a rock horizontally

with a speed of 6.75 m/s. The rock falls a distance of 1.20 m

and lands a horizontal distance of 8.95 m from the astronaut.

What is the acceleration due to gravity on Zircon?


51/68

For The Adventurous

Here are some more advanced equations.

They are not given on the equation sheet

You are free to use them if they are memorized

You can have programs stored in your calc

Unlikely you will need them, but


52/68

Vertical Position Equation

vo hx

yVertical Position as a function

of vo, , and horizontal position

sin, oyo vv 2

,21 gttvy yo

2,

21 tatvx xxo

cos, oxo vv

cosov

xt

2

cos2

1

cossin

oo

ov

x

gv

x

vy

22

2

cos2tan

ov

xgxy

0


53/68

Maximum Height

gtvv oy sin 0

g

vt otop

sin

vo h

Equation for the maximum

Height of a projectile.

At max height,

vy = 0 2

2

1sin tgtvy o

2

sin

2

1sinsin

g

vg

g

vvy ooo

g

vy o

2

sin22

max


54/68

Range Equation

tvx o cos 22

1sin tgtvy o

g

vt o

sin2

g

vvx oo

sin2cos

uuu cossin22sin

2sin2

g

vx o

cossin2

2

g

vx o

0

vo

d


55/68


Section 4:

Relative Veloc ity


56/68

Relative Motion Problems

Describe motion of an object from some fixed point

Relative motion problems are difficult to do unless one applies

vector addition concepts

Define a vector for an object velocity relative to the fluid, and

another for the velocity of the fluid relative to the observer.

Add two vectors together.


57/68

TGPTPG vvv

velocity of person

relative to the ground

velocity of person

relative to the train

velocity of train

relative to the ground


58/68

The engine of a boat drives it across a river that is 1800m wide.

The velocity of the boat relative to the water is 4.0m/s directed

perpendicular to the current. The velocity of the water relative

to the shore is 2.0m/s.

(a) What is the velocity of the

boat relative to the shore?

(b) How long does it take for

the boat to cross the river?

Example 11 Crossing a River


59/68

22

sm0.2sm0.4 BSv

WSBWBS vvv

0.2

0.4tan

1

x vBW vWS vBS t

1800m 4.0m/s 2.0m/s ? ?

22

WSBWBS vvv

sm5.4BSv

WS

BW

v

v1tan

63

t


60/68

sm4.0

m1800t

x vBW vWS t

1800m 4.0m/s 2.0m/s ?

BWv

xt

s450

2

2

1attvx BW

0


61/68

Question #14

You are trying to cross a river that flows due south

with a strong current. You start out in your motorboat

on the east bank desiring to reach the west bank

directly west from your starting point. You shouldhead your motorboat

a) due west.

b) due north.c) in a southwesterly direction.

d) in a northwesterly direction.


62/68

Question #15

At an air show, three planes are flying horizontally due

east. The velocity of plane A relative to plane B is vAB;

the velocity of plane A relative to plane C is vAC; and

the velocity of plane B relative to plane C is vBC.Determine vABif vAC= +10 m/s and vBC= +20 m/s?

a) 10 m/s b) +10 m/s

c) 20 m/s d) +20 m/s

e) zero m/s


63/68

Question #16

A train is traveling due east at a speed of 26.8 m/s relative tothe ground. A passenger is walking toward the front of thetrain at a speed of 1.7 m/s relative to the train. Directlyoverhead the train is a plane flying horizontally due west at aspeed of 257.0 m/s relative to the ground. What is thehorizontal component of the velocity of the airplane withrespect to the passenger on the train?

a) 258.7 m/s, due west b) 285.5 m/s, due west

c) 226.8 m/s, due west d) 231.9 m/s, due west

e) 257.0 m/s, due west


64/68

Question #17

Sailors are throwing a football on the deck of an aircraft carrier as it is sailing with aconstant velocity due east. Sailor A is standing on the west side of the flight deckwhile sailor B is standing on the east side. Sailors on the deck of another aircraftcarrier that is stationary are watching the football as it is being tossed back andforth as the first carrier passes. Assume that sailors A and B throw the footballwith the same initial speed at the same launch angle with respect to the horizontal,

do the sailors on the stationary carrier see the football follow the same parabolictrajectory as the ball goes east to west as it does when it goes west to east?

a) Yes, to the stationary sailors, the trajectory the ball follows is the same whetherit is traveling west to east or east to west.

b) No, to the stationary sailors, the length of the trajectory appears shorter as ittravels west to east than when it travels east to west.

c) No, to the stationary sailors, the ball appears to be in the air for a much longer

time when it is traveling west to east than when it travels east to west.d) No, to the stationary sailors, the length of the trajectory appears longer as ittravels west to east than when it travels east to west.

e) No, to the stationary sailors, the ball appears to be in the air for a much shortertime when it is traveling west to east than when it travels east to west.


65/68

Question #18Cars A and B are moving away from each other as car A movesdue north at 25 m/s with respect to the ground and car B movesdue south at 15 m/s with respect to the ground. What are thevelocities of the other car according to the two drivers?

a) Car A is moving due north at 25 m/s; and car B is moving due

south at 15 m/s.b) Car A is moving due south at 25 m/s; and car B is moving duenorth at 15 m/s.

c) Car A is moving due north at 40 m/s; and car B is moving duesouth at 40 m/s.

d) Car A is moving due south at 40 m/s and car B is moving duenorth at 40 m/s.

e) Car A is moving due north at 15 m/s and car B is moving duesouth at 25 m/s.


66/68

Question #19John always paddles his canoe at constant speed vwith respectto the still water of a river. One day, the river current was duewest and was moving at a constant speed that was close to vwith respect to that of still water. John decided to see whethermaking a round trip across the river and back, a north-south trip,

would be faster than making a round trip an equal distance east-west. What was the result of Johns test?

a) The time for the north-south trip was greater than the time forthe east-west trip.

b) The time for the north-south trip was less than the time for theeast-west trip.

c) The time for the north-south trip was equal to the time for theeast-west trip.

d) One cannot tell because the exact speed of the river withrespect to still water is not given.


67/68

Question #20

A boat attempts to cross a river. The boats speed with

respect to the water is 12.0 m/s. The speed of the river

current with respect to the river bank is 6.0 m/s. At what

angle should the boat be directed so that it crosses the

river to a point directly across from its starting point?

BW

WS

v

v1sin

6.0m/s

sm

sm

12

0.6sin 1


68/68

ch3twodimensionalkinematicsonline-101013171228-phpapp02

Documents