Motion in One DimensionMotion in One Dimension

Golden Valley HS Physics Mr. Golden Valley HS Physics Mr. CampbellCampbell

2-1 Displacement and Velocity2-1 Displacement and Velocity

MotionMotionMotion is what happens all around us.Motion is what happens all around us.

Different directions and different speeds.Different directions and different speeds.

It requires a special effort to analyze motion as a It requires a special effort to analyze motion as a physicist does.physicist does.

One-dimensional motion is the simplest form of One-dimensional motion is the simplest form of motionmotion

One way to simplify motion is to consider it only in One way to simplify motion is to consider it only in oneone direction.direction.

Think of a commuter train on a straight track.Think of a commuter train on a straight track.

It can only move It can only move forwardforward or backward. or backward.

Motion takes place over time and Motion takes place over time and depends upon the frame of referencedepends upon the frame of reference

Describing the motion of a train is simple.Describing the motion of a train is simple.

Earth is Earth is spinningspinning on an axis, moving around the on an axis, moving around the sunsun and and our sun is moving inside of a our sun is moving inside of a galaxygalaxy..

To help simplify what we are studying we can use a To help simplify what we are studying we can use a frame of frame of referencereference. It is a point that stays fixed.. It is a point that stays fixed.

The gecko’s motion is measured along the x axis using The gecko’s motion is measured along the x axis using centimeters.centimeters.

Motion

Displacement Displacement

Displacement is a change in positionDisplacement is a change in positionThe initial position to the final position is called the The initial position to the final position is called the

displacementdisplacement of the object. of the object.

The gecko starts at xThe gecko starts at xii and moves to x and moves to xff..

The Greek letter (The Greek letter (ΔΔ) “) “deltadelta” shows change.” shows change.

Displacement can also be shown Displacement can also be shown verticallyvertically

with the y axis.with the y axis.

if xxx

Displacement

Displacement is not always equal to the Displacement is not always equal to the distance traveleddistance traveled

If the gecko runs up the tree as shown, the If the gecko runs up the tree as shown, the changechange in displacement is simply y in displacement is simply yff – y – yii or 60 cm. or 60 cm.

If the gecko moves back down to 50 cm, If the gecko moves back down to 50 cm,

50 cm – 20 cm = 30 cm of total 50 cm – 20 cm = 30 cm of total displacementdisplacement

(based on the starting point). (based on the starting point).

If the gecko returns to its starting point its total If the gecko returns to its starting point its total displacement will be “0”.displacement will be “0”.

Displacement can be positive or negative Displacement can be positive or negative

In the below example there are only In the below example there are only twotwo directions our directions our object can move.object can move.

Right will be considered Right will be considered easteast or positive x. Left will be west or positive x. Left will be west or or negativenegative x. x.

Up with be Up with be northnorth(positive y) down will be south(negative y)(positive y) down will be south(negative y)

VelocityVelocity

Where an object starts and where it stops does not Where an object starts and where it stops does not completely describe its motion.completely describe its motion.

The ground you are standing on may be moving to the The ground you are standing on may be moving to the left at 2 cm per year (plate tectonics).left at 2 cm per year (plate tectonics).

If there is an earthquake, we could move 10 cm in a If there is an earthquake, we could move 10 cm in a secondsecond! !

Average velocity is displacement divided by the Average velocity is displacement divided by the time intervaltime interval

Average velocityAverage velocity is the displacement divided is the displacement divided by the by the timetime interval. interval.

The S.I. unit is meters per The S.I. unit is meters per secondsecond (m/s) (m/s)

This value is an This value is an averageaverage..

You may have stopped for a while, traveled You may have stopped for a while, traveled slower or faster for a bit.slower or faster for a bit.

if

ifavg tt

xx

t

xv

Velocity

Practice A: Practice A: Average Velocity and DisplacementAverage Velocity and Displacement

During a race on level ground, Andra runs During a race on level ground, Andra runs with an average velocity of 6.02 m/s to the with an average velocity of 6.02 m/s to the east. What is Andra’s displacement after east. What is Andra’s displacement after 137 s?137 s?

answeranswer

t

xvavg

tvx avg

825 m to the east

Velocity is not the same as speedVelocity is not the same as speedSpeed and velocity sometimes are used interchangeably.Speed and velocity sometimes are used interchangeably.

In Physics they are different.In Physics they are different.

VelocityVelocity describes the magnitude of that motion and describes the magnitude of that motion and directiondirection..

Speed Speed however, has however, has nono direction only direction only magnitudemagnitude..

Velocity can be interpreted graphically Velocity can be interpreted graphically Velocity can be plotted on a graph.Velocity can be plotted on a graph.

Position is normally plotted on the Position is normally plotted on the verticalvertical axis and time axis and time on the horizontal.on the horizontal.

If we have constant velocity, we will have a If we have constant velocity, we will have a straightstraight line line

This graph shows…This graph shows…

– Object 1:Object 1: positive slope = positive velocity positive slope = positive velocity– Object 2:Object 2: zero slope = zero velocity zero slope = zero velocity – Object 3:Object 3: negative slope = negative velocity negative slope = negative velocity

Sign Conventions

Instantaneous velocity may not be the same as Instantaneous velocity may not be the same as average velocityaverage velocity

This graph show This graph show changingchanging velocity velocity with time.with time.

This causes a This causes a curvedcurved line on our line on our graph.graph.

Any single point on this graph Any single point on this graph shows shows instantaneous velocityinstantaneous velocity..

The instantaneous velocity at a The instantaneous velocity at a given time can be determined by given time can be determined by measuring the slope of the line that measuring the slope of the line that is is tangenttangent to that point on the to that point on the position-versus-time graph.position-versus-time graph.

The slope and shape of a graph of acceleration describe the object’s motion.

2.1 Questions2.1 Questions

1. What is a frame of reference?1. What is a frame of reference?__________________________________________2. The Greek letter (2. The Greek letter (ΔΔ) “delta” shows ________) “delta” shows ________3. T / F Displacement is always equal to distance 3. T / F Displacement is always equal to distance traveled.traveled.4. Average velocity is the displacement divided by 4. Average velocity is the displacement divided by ____________________5. Speed has no direction, only __________5. Speed has no direction, only __________

The part that does not movechange

False

timemagnitude

Sec 2-2 AccelerationSec 2-2 Acceleration

Changes in VelocityChanges in VelocityAcceleration is the rate of change of velocity with Acceleration is the rate of change of velocity with respect to timerespect to time

A bus approaches a stop and begins to apply the brakes.A bus approaches a stop and begins to apply the brakes.

The speed changes from 9.0 m/s to 0 m/s in over a period The speed changes from 9.0 m/s to 0 m/s in over a period of 5 seconds.of 5 seconds.

If a dog were to run out and the bus stopped in 1.5 s, If a dog were to run out and the bus stopped in 1.5 s, these two stops would be very different for the passengers.these two stops would be very different for the passengers.

The rate of change of velocity in a given time is called The rate of change of velocity in a given time is called accelerationacceleration..

The magnitude of average acceleration is found by The magnitude of average acceleration is found by dividing the total change in velocity by the dividing the total change in velocity by the timetime it takes. it takes.

The S.I. unit is the The S.I. unit is the metermeter per second per per second per secondsecond (m/s (m/s22) ) or how much the velocity changes each second.or how much the velocity changes each second.

if

ifavg tt

vv

t

va

Changes in Velocity

Practice B: Practice B: Average AccelerationAverage Acceleration

A shuttle bus slows down with an average acceleration A shuttle bus slows down with an average acceleration of -1.8 m/sof -1.8 m/s22. How long does it take the bus to slow from . How long does it take the bus to slow from 9.0 m/s to a complete stop?9.0 m/s to a complete stop?

Given: Given: vvii = = 9.00 m/s9.00 m/s vvff = = 0 m/s0 m/s aaavgavg = = -1.8 m/s -1.8 m/s22

AnswerAnswer

if

ifavg tt

vv

t

va

avg

if

avg a

vv

a

vt

5.0 s

Acceleration has direction and magnitudeAcceleration has direction and magnitudeVelocity is positive when a train moves to the Velocity is positive when a train moves to the rightright..

If the velocity increase as it travels in this direction, the final If the velocity increase as it travels in this direction, the final velocity will be greater than when it started.velocity will be greater than when it started.

When change in “When change in “vv” is positive, acceleration is also positive.” is positive, acceleration is also positive.

On long trips the train may stay at constant velocity for a while. On long trips the train may stay at constant velocity for a while. If velocity is not changing If velocity is not changing ΔΔv = v = 0 m/s. Therefore acceleration is 0 m/s. Therefore acceleration is zerozero..

When the train slows down, its velocity When the train slows down, its velocity

is still positive but changing.is still positive but changing.

Because initial velocity is greater than Because initial velocity is greater than

final. Acceleration is final. Acceleration is negativenegative..

The slope and shape of the graph describe the The slope and shape of the graph describe the object’s motionobject’s motion

Looking at this graph, we can see the speed of the train Looking at this graph, we can see the speed of the train is increasing over time.is increasing over time.

““A” shows we have both positive direction and A” shows we have both positive direction and acceleration.acceleration.

““B” shows positive direction and constant speed – no B” shows positive direction and constant speed – no acceleration.acceleration.

““C” shows continued positive C” shows continued positive

direction but negative acceleration.direction but negative acceleration.

If the train were moving backward (negative direction) If the train were moving backward (negative direction) and speeding up, we could have and speeding up, we could have negativenegative acceleration acceleration also.also.

When looking at this table, take into consideration which When looking at this table, take into consideration which way the train is moving.way the train is moving.

If If vvii is (-) then the train is moving backward. is (-) then the train is moving backward.

Motion with Constant AccelerationMotion with Constant AccelerationThis photo was taken with a strobe light flashing This photo was taken with a strobe light flashing

every 0.1 second.every 0.1 second.

Gravity Gravity acceleratesaccelerates things as they fall. This object things as they fall. This object is undergoing constant acceleration.is undergoing constant acceleration.

Since its speed changes with time, it covers a Since its speed changes with time, it covers a greater greater distancedistance in each constant time interval. in each constant time interval.

Displacement depends on acceleration, initial Displacement depends on acceleration, initial velocity and timevelocity and time

This graph shows an object with This graph shows an object with

constant acceleration is going constant acceleration is going

faster with time.faster with time.

With constant acceleration, average velocity is equal to the With constant acceleration, average velocity is equal to the average between initial and average between initial and finalfinal velocity. velocity.

To find displacement, we substitute To find displacement, we substitute xx and and tt into the equation. into the equation. This equation makes finding This equation makes finding displacementdisplacement easier. easier.

tvvx fi 2

1

Practice C: Practice C: Displacement with Constant AccelerationDisplacement with Constant Acceleration

A racing car reaches a speed of 42 m/s. It then A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its begins a uniform negative acceleration, using its parachute and braking system, and comes to a parachute and braking system, and comes to a rest 5.5 seconds later.rest 5.5 seconds later.

Find the distance that the car travels during Find the distance that the car travels during braking.braking.

AnswerAnswer

tvvx fi 2

1

120 m

Final velocity depends on initial velocity, Final velocity depends on initial velocity, acceleration and timeacceleration and time

You can use this equation to find final velocity of an object You can use this equation to find final velocity of an object after it has accelerated at a after it has accelerated at a constantconstant rate for any time rate for any time interval.interval.

For For displacementdisplacement of an object use… of an object use…

This can be used to find displacement and a certain This can be used to find displacement and a certain speedspeed after a displacement has been reached.after a displacement has been reached.

tavv if

22

1tatvx i

Practice D: Practice D: Velocity and displacement with uniform Velocity and displacement with uniform accelerationacceleration

A plane starting from rest at one end of a runway A plane starting from rest at one end of a runway undergoes a uniform acceleration of 4.8 m/sundergoes a uniform acceleration of 4.8 m/s22 for for 15 s before takeoff.15 s before takeoff.

What is its speed at takeoff?What is its speed at takeoff?

AnswerAnswer

How long must the runway be for the plane to be How long must the runway be for the plane to be able to take off?able to take off?

AnswerAnswer

tavv if

22

1tatvx i

72 m/s

540 m

Final velocity depends on initial velocity, Final velocity depends on initial velocity, acceleration and displacementacceleration and displacement

So far our equations for acceleration have required a So far our equations for acceleration have required a time interval.time interval.

Now we will rearrange the equation, substituting time to Now we will rearrange the equation, substituting time to find find distancedistance or displacement. or displacement.

This new equation gives us final velocity after a This new equation gives us final velocity after a displacement…displacement…

When using tis equation, you must take the When using tis equation, you must take the squaresquare root root of the right side to find final velocity. of the right side to find final velocity.

xavv if 222

Practice E: Practice E: Final velocity after any displacementFinal velocity after any displacement

A person pushing a stroller starts from rest, uniformly A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500 m/saccelerating at a rate of 0.500 m/s22..

What is the velocity of the stroller after it has traveled What is the velocity of the stroller after it has traveled 4.75 m?4.75 m?

Given: Given: vvii = = 0 m/s 0 m/s a = a = 0.500 m/s0.500 m/s22 ΔΔx =x = 4.75 m 4.75 m

vvff = = ??

AnswerAnswer

xavv if 222

+2.18 m/s

xavv if 22

The equations you have just used are summarized The equations you have just used are summarized below.below.

They can also be referenced on P. They can also be referenced on P. 5858 of you textbook. of you textbook.

If the object starts from If the object starts from restrest ( (vvi i = = 0) you can you the 0) you can you the

formulas on the right.formulas on the right.

2.2 Questions2.2 Questions

1. Acceleration is the rate of change of _________ 1. Acceleration is the rate of change of _________ with respect to time.with respect to time.2. Acceleration has direction and __________2. Acceleration has direction and __________3. When a train slows down, its velocity is still 3. When a train slows down, its velocity is still __________ but changing.__________ but changing.4. Gravity 4. Gravity ____________________ things as they fall. things as they fall.5. T / F If we have final velocity, initial velocity and 5. T / F If we have final velocity, initial velocity and acceleration, we can calculate time.acceleration, we can calculate time.

velocity

magnitude

positiveaccelerates

True

Sec 2-3 Falling ObjectsSec 2-3 Falling Objects

Free FallFree FallIn 1971, astronaut David Scott showed In 1971, astronaut David Scott showed

on the moon (a vacuum), a hammer and on the moon (a vacuum), a hammer and a a featherfeather will fall at the same rate in the will fall at the same rate in the absence of air absence of air resistanceresistance..

Both objects hit the moon’s surface at Both objects hit the moon’s surface at the the samesame time. time.

Freely falling bodies undergo constant Freely falling bodies undergo constant accelerationacceleration

In a In a vacuumvacuum chamber, any two objects will fall exactly at chamber, any two objects will fall exactly at the same rate.the same rate.

In the photo you can see the time interval is the same In the photo you can see the time interval is the same but the but the displacementdisplacement keeps changing. keeps changing.

This shows the objects are This shows the objects are acceleratingaccelerating..

This constant acceleration is called This constant acceleration is called free fallfree fall..

The magnitude of acceleration due to gravity (g) is -9.81 The magnitude of acceleration due to gravity (g) is -9.81 m/sm/s22..

This acceleration is directed downward, therefore it is This acceleration is directed downward, therefore it is negativenegative..

What goes up must come down

Freely fall constant acceleration

Acceleration is constant during Acceleration is constant during

upward and downward motionupward and downward motionThe object shown in this strobe light photo goes up, The object shown in this strobe light photo goes up,

slows, changes direction and falls.slows, changes direction and falls.

The acceleration on the object is The acceleration on the object is negativenegative the entire time. the entire time. Even when the object is going up, it experiences a force Even when the object is going up, it experiences a force ((gravitygravity) pulling it down.) pulling it down.

If it were If it were notnot for this negative acceleration (gravity), the for this negative acceleration (gravity), the ball would travel up continuously.ball would travel up continuously.

This graph shows gravity’s negative acceleration plotted This graph shows gravity’s negative acceleration plotted with with changchange in velocity.e in velocity.

Because acceleration is negative, it continually slopes Because acceleration is negative, it continually slopes down to the down to the rightright..

Objects thrown upward same acceleration as falling.

Freely falling objects always have the same Freely falling objects always have the same downward accelerationdownward acceleration

It may seem confusing to think an object moving up is It may seem confusing to think an object moving up is undergoing a negative acceleration.undergoing a negative acceleration.

When an object is moving When an object is moving downdown it has negative it has negative velocityvelocity and negative and negative accelerationacceleration, therefore it is going faster with , therefore it is going faster with time.time.

When it is thrown up, it has When it is thrown up, it has positivepositive velocity but negative velocity but negative acceleration so it is acceleration so it is slowingslowing down. down.

Practice F: Practice F: Falling ObjectFalling Object

Jason hits a volleyball so that it moves with Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. an initial velocity of 6.0 m/s straight upward.

If the volleyball starts from 2.0 m above the If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it floor, how long will it be in the air before it strikes the floor?strikes the floor?

Given: Given: vvii = = +6.0 m/s +6.0 m/s aagg = = -9.81 m/s-9.81 m/s22 ΔΔy = y = -2.0 m-2.0 m

AnswerAnswer When ball hits the floor, next When ball hits the floor, next yavv if 222yavv if 22

-8.7 m/s

……to figure out the overall time the ball was in to figure out the overall time the ball was in the air, we use…the air, we use…

AnswerAnswer

This the time it took the ball to go up, go down This the time it took the ball to go up, go down and then another 2 meters.and then another 2 meters.

tavv if avvt if

1.50 s

2.3 Questions2.3 Questions

1. In the absence of air, all objects fall at the same 1. In the absence of air, all objects fall at the same ____________2. Free falling objects undergo constant ________2. Free falling objects undergo constant ________3. T / F Acceleration is constant during upward 3. T / F Acceleration is constant during upward and downward motion.and downward motion.4. Objects thrown up have positive velocity but 4. Objects thrown up have positive velocity but negative acceleration, this causes them to ______ negative acceleration, this causes them to ______ down.down.

rateacceleration

True

slow

EndEnd