Chapter 2Motion in One Dimension

Classical Physics Refers to physics before the 1900’s

▪ Kinematics/Dynamics▪ Electromagnetism▪ Thermodynamics

Applies to everyday phenomena Modern Physics

Refers to post 1900’s physics▪ Quantum Mechanics▪ Relativity▪ Nuclear Physics

Applies to very small/big phenomena

Kinematics (special branch of mechanics) Study of motion Irrespective of causes (dynamics) Three most important concepts

▪ Displacement▪ Velocity▪ Acceleration

Assumptions Ideal Particle

▪ Classical physics concept▪ Point-like object, no size▪ Real particles have size, charge, spin

Time is absolute▪ Independent of position or velocity▪ Relativity says time is not absolute!

▪ Twin Paradox

Define frame of reference Coordinate system

▪ Rectangular Simplest case

▪ 1-Dimensional▪ Can extrapolate to other dimensions (independent)

Only motion along the straight line is possible

1 2 3 4 5 6 7 8 9 X

Allows keep track of direction Positive direction Negative direction

When you apply equations for the motion of bodies it is very important to keep track of the direction of motion Negative and Positive values Affect result

The displacement of a moving object moving along the x-axis is defined as the change in the position of the object,

Δx = xf – xi

Where xi is the initial position and xf is the final position

First DisplacementΔx1 = xf – xi = 52 – 30 = 22 mSecond DisplacementΔx2 = xf – xi = 38 – 52 = -14 m can have negative displacement – backwards

Vector – both magnitude and direction Ex: Velocity – magnitude (speed) and

direction▪ 55 mi/hr North East

▪ 45º North of East

Ex: Electric Field – magnitude and direction▪ Electron

▪ Perpendicular to surface

Scalar – magnitude only Ex: temperature, density, mass, volume

NE

E

N

Think of a vector as an arrow (pointing to some direction), and its magnitude as the length of the arrow (always positive, independent of the direction of the arrow).

You and your friend go for a You and your friend go for a

walk to the park. On the way, walk to the park. On the way,

your friend decides to text while your friend decides to text while

walking and wanders off and takes walking and wanders off and takes

a few side trips by dodging cars a few side trips by dodging cars

and falling off a bridge. When you and falling off a bridge. When you

both arrive at the park, do you both arrive at the park, do you

and your friend have the same and your friend have the same

displacement?displacement?

1) yes

2) no

ConcepTest ConcepTest Texting While WalkingTexting While Walking

You and your dog go for a walk to You and your dog go for a walk to

the park. On the way, your dog the park. On the way, your dog

takes many side trips to chase takes many side trips to chase

squirrels or examine fire hydrants. squirrels or examine fire hydrants.

When you arrive at the park, do When you arrive at the park, do

you and your dog have the same you and your dog have the same

displacement?displacement?

1) yes

2) no

Yes, you have the same displacement. Since

you and your friend had the same initial position

and the same final position, then you have (by

definition) the same displacement.

ConcepTest ConcepTest Texting While WalkingTexting While Walking

Follow-up:Follow-up: Have you and your friend traveled the same Have you and your friend traveled the same distance?distance?

The average speed of an object is given by:

Average speed = total distance / total times = d / t > 0 always

Speeds (m/s):▪ Light 3 x 108 ▪ Sound 343▪ Person 10▪ Fastest Car 110▪ Continent 10-8

Speed and motion are relative Depends on frame of reference

SunEarth

Orbit speed of earth around the sun - 29.7 km/s

A person running on earth - 5 m/s

The average velocity v during a time interval Δt is the displacement Δx divided by Δt :

v = — = ——

One dimension, straight line motion The average velocity is equal to the

slope of the straight line joining the initial and final points on a graph of the position vs. time

tf - ti

xf - xi

Δt

Δx(m/s)

The figure below shows the graphical interpretation of the average velocity (A to B), in the case of an object moving with a variable velocity:

Example:A motorist drives north for 35.0 minutes at 85.0 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.00 h. (a) What is his total displacement? (b) What is his average velocity?

The slope of the line tangent to the position vs. time curve at some point is equal to the instantaneous velocity at that time.

t

x

v = lim ——Δx

ΔtΔx -> 0

Position vs. Time graphs

                       

  

                       

   

An object moving with a constant velocity will have a graph that is a straight line

An object moving with a non-constant velocity will have a graph that is a curved line

Position vs. Time Time is always moving forward

Example:

a) What is the velocity from O to A?

b) What is the velocity from A to B?

c) What is the velocity from O to C?

d) What is the instantaneous velocity at t=2?

Note: Average velocity does not necessarily have the same magnitude as average speed

Average speed = ————————

Average velocity = ————————

Distance Travelled

Time

Time

Displacement

x1

x3

t1 t2s12 = —————— ≠ 0

v12 = —————— = 0

x2

x2 - x1

t2 - t1 x2 - x1

t2 - t1

1. A yellow car is heading East at 100 km/h and a red car is going North at 100 km/h. Do they have the same speed? Do they have the same velocity?

2. A 16-lb bowling ball in a bowling alley in Folsom Lanes heads due north at 10 m/s. At the same time, a purple 8-lb ball heads due north at 10 m/s in an alley in San Francisco. Do they have the same velocity?

• The average acceleration v of an object with a change of velocity Δv during a time interval Δt :

a = — = ——

• The instantaneous acceleration of an object at a certain time equals the slope of a velocity vs. time graph at that instant.

tf - ti

vf - vi

Δt

Δv(m/s2)

Average Acceleration

Acceleration is a vector quantity It has both a magnitude and a direction.

Positive or negative to indicate direction (of acceleration!)

Acceleration is positive when the velocity increases in the positive direction

Furthermore, the velocity increases when the acceleration and the velocity point in the same direction, it decreases when they are pointing in opposite directions

Positive Acceleration

Negative Acceleration

Positive Acceleration (Negative direction)

If the velocity of a car is non-If the velocity of a car is non-

zero (zero (v v 00), can the ), can the

acceleration of the car be zero?acceleration of the car be zero?

1) yes

2) no

3)

depends

on the

velocity

Sure it can! An object moving with constantconstant

velocityvelocity has a non-zero velocity, but it has

zerozero accelerationacceleration since the velocity is not

changing.

If the velocity of a car is non-If the velocity of a car is non-

zero (zero (v v 00), can the acceleration ), can the acceleration

of the car be zero?of the car be zero?

1) yes

2) no

3) depends on the

velocity

Velocity vs. Time graphs

Velocity vs. Time graphs

Velocity vs. Time graphs

Example:A car traveling in a straight line has a velocity of + 5.0 m/s at some instant. After 4.0 s, its velocity is + 8.0 m/s. What is the car’s average acceleration during the 4.0-s time interval?

Useful to look at situations when the acceleration is constant

▪ Velocity is changing Motion in a straight line

If the acceleration is constant, then the average acceleration is equal to the acceleration itself: a = constant => <a> = a = constant

Applies to gravity and other situations as well

Important (and useful) equations with straight line, uniform acceleration

1. v = v0 + at2. Δx = vt = ½(v0 + v)t3. Δx = v0t + ½ at2

4. v2= v02 + 2a Δx

ALL you need to solve problems of kinematics with constant acceleration in 1 dimension

Examples

The most important example of 1d motion with uniform acceleration is gravity

A free-falling object is an object falling under the influence of gravity alone

All objects fall near the earth’s surface with a constant acceleration, g

g = 9.8 m/s2

g is always directed downwardAll objects, regardless of mass, free-

fall at the same acceleration

An example of free fall is the collapse of gas in the formation of stars

Gravitational Force Dependent on where we are on Earth, because

the Earth is not a perfect sphere and because the presence of large masses (e.g. mountains) also affects the local gravitational force

For now, just consider the force of gravity as a constant force pulling any object towards the center of the Earth, and therefore perpendicular to the ground and towards the ground.▪ Approximate constant acceleration pointing to the ground

Feather and Hammerhttp://www.youtube.com/watch?v=KDp1tiUsZw8

Falling objectshttp://www.youtube.com/watch?v=_XJcZ-KoL9o

ConcepTest 2.9bConcepTest 2.9b Free Fall II Free Fall II

Alice and Bill are at the top of a Alice and Bill are at the top of a

building. Alice throws her ball building. Alice throws her ball

downward. Bill simply drops downward. Bill simply drops

his ball. Which ball has the his ball. Which ball has the

greater acceleration just after greater acceleration just after

release?release?

1) Alice’s ball 1) Alice’s ball

2) it depends on how hard 2) it depends on how hard the ball was thrownthe ball was thrown

3) neither -- they both have 3) neither -- they both have the same accelerationthe same acceleration

4) Bill’s ball4) Bill’s ball

vv

00

BillBillAliceAlice

vv

AA

vv

BB

Both balls are in free fall once they are

released, therefore they both feel the

acceleration due to gravity (g). This

acceleration is independent of the initial

velocity of the ball.

ConcepTest 2.9bConcepTest 2.9b Free Fall II Free Fall II

Alice and Bill are at the top of a Alice and Bill are at the top of a

building. Alice throws her ball building. Alice throws her ball

downward. Bill simply drops downward. Bill simply drops

his ball. Which ball has the his ball. Which ball has the

greater acceleration just after greater acceleration just after

release?release?

1) Alice’s ball 1) Alice’s ball

2) it depends on how hard 2) it depends on how hard the ball was thrownthe ball was thrown

3) neither -- they both have 3) neither -- they both have the same accelerationthe same acceleration

4) Bill’s ball4) Bill’s ball

vv

00

BillBillAliceAlice

vv

AA

vv

BBFollow-up:Follow-up: Which one has the greater velocity when they hit Which one has the greater velocity when they hit the ground?the ground?

All objects, regardless of mass, free-fall at the same acceleration More detail in chapters on force and mass Free-falling objects do not encounter air

resistance We can use the same four important

equations from before (because g is constant acceleration) but change x direction to y direction Substitute a = -g

Substitute a = -g, y-direction

1. v = v0 - gt2. Δy = vt = ½(v0 + v)t3. Δy = v0t - ½ gt2

4. v2= v02 - 2g Δy

Imagine dropping an object, and measuring

how fast it’s moving over consecutive 1

second intervals http://www.youtube.com/watch?v=xQ4znShl

K5A The vertical component of velocity is

changing by 9.8 m/s in each second,

downwards Let’s approximate this acceleration as 10

m/s2

TimeInterval

Acceleration(m/s2

down)

Vel. at end of interval(m/s down)

0 – 1 s

10 10

1 – 2 s

10 20

2 – 3 s

10 30

3 – 4 s

10 40

4 – 5 s

10 50

Starting from rest, then letting go.After an interval t, the velocity changes by an amount at, so that

vfinal = vinitial + at

How fast was it going at the end of 3 sec?vinitial was 20 m/s after 2 seca was 10 m/s (as always)t was 1 sec (interval)

vfinal = 20 m/s + 10 m/s2 1 s = 30 m/s

You throw a ball upward with an You throw a ball upward with an

initial speed of 10 m/s. initial speed of 10 m/s.

Assuming that there is no air Assuming that there is no air

resistance, what is its speed resistance, what is its speed

when it returns to you?when it returns to you?

1) more than 10 m/s 1) more than 10 m/s

2) 10 m/s2) 10 m/s

3) less than 10 m/s3) less than 10 m/s

4) zero4) zero

5) need more information5) need more information

The ball is slowing down on the way up due to

gravity. Eventually it stops. Then it accelerates

downward due to gravity (again). Since a = g on

the way up and on the way down, the ball reaches

the same speed when it gets back to you as it had

when it left.

You throw a ball upward with an You throw a ball upward with an

initial speed of 10 m/s. initial speed of 10 m/s.

Assuming that there is no air Assuming that there is no air

resistance, what is its speed resistance, what is its speed

when it returns to you?when it returns to you?

1) more than 10 m/s 1) more than 10 m/s

2) 10 m/s2) 10 m/s

3) less than 10 m/s3) less than 10 m/s

4) zero4) zero

5) need more information5) need more information

a) Find the time when the stone reaches its maximum height.

b) Determine the stone’s maximum height.

c) Find the time the stone takes to return to its final position and find the velocity of the stone at that time.

d) Find the time required for the stone to reach the ground.

Example