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EVERYTHING MOVES!!!!! THERE IS NO APSOLUTE REST!!!!

The Sea Serpent

MOTION

Even while sitting in the classroom appearing motionless, you are moving very fast.

• 0.4 km/s (0.25 mi/s) rotating around the center of the Earth

• 30 km/s relative to the Sun

• even faster relative to the center of our galaxy (The Sun orbits the center of the Milky Way at about 250 km/s and it takes about 220 million years to complete an orbit. )

• together with the whole galaxy moving away from the center of the Universe at huge speed. Measurements, confirmed by the Cosmic Background Explorer satellite in 1989 and 1990, suggest that our galaxy and its neighbors, are moving at 600 km/s (1.34 million mi/h) in the direction of the constellation Hydra.).

When we discuss the motion of something, we describe its motion relative to something else.

When you say that you drove your car at speed of 50 mi/h, of course you mean relative to the road or with respect to the surface of the Earth. You are actually using coordinate system without knowing it.

A frame of referenceis a perspective from which a system is observed together with a coordinate system used to describe motion of that system.

One of the important problem problems in Physics is this:

if a any given instant in time we know the positions and velocities of all the particles that make up a particular system can we predict the future position and velocities of all the particles?

If we can do it then we can:

predict solar eclipses, put satellites into orbit, find out how the

position of a swing varies with time and find out where a soccer ball ends up when struck by a foot.

Classical Mechanics: - Study of the motion of macroscopic objects and related concepts of force and energy

Kinematics – is concerned with the description of how objects

move; their motion is described in terms of displacement,

velocity, and acceleration

Dynamics– explains why objects change the state of the motion

(velocity) as they do; explains motion and causes of changes using

concepts of force and energy.

The movement of an object through space can be quite complex.

There can be internal motions, rotations, vibrations, etc…

This is the combination of rotation (around its center of mass) and the motion along a line - parabola.

If we treat the hammer as a particlethe only motion is translational motion (along a line) through space.

Our objects will be represented as point objects (particles) so they move through space without rotation.

We’ll be neglecting all factors such as the shape, size, etc. that make the problem too difficult for now.

Simplest motion: motion of a particle along a line – called:

translational motion or one-dimensional (1-D) motion.

displacement

Q

path;

length is distance traveled

Displacement

- is the shortest distance in a given direction.
- The displacement tells us how far an object is from
- its starting position and in what direction

(it tells us how the object is displaced)

Distance between final and initial point

Example:

1) x1 = 7 m, x2 = 16 mx3 = 12 m∆x = 5 m

Representation of displacement in a coordinate system

“+” direction

2) x1 = 7 m, x2 = 2 m∆ x = - 5 m

“ –” direction

A racing car travels round a circular track of radius 100 m.

The car starts at O. When it has travelled to P its displacement as measured from O is

A 100 m due East

B 100 m due West

C 100 √2 m South East

D 100 √2 m South West

Each physical quantity will be either a scalar or a vector.

Scalaris a quantity that is completely specified by a positive or negative number with appropriate units.

Temperature, length, mass, time, speed, …

Vector must be specified by both magnitude

(number and unit) and direction.

Displacement, velocity, force,…

Scalar Vector

distance - 50 km displacement: 50 km, E

speed - 70 km s-1 velocity: 70 km s-1, S-W

Scalars obey the rules of ordinary algebra:

2 kg of potato + 2 kg of potato = 4 kg of potato

Vectors obey the rules of vectors’ algebra:

The sum of two vectors depends on their directions.

Average and Instantaneous Velocity

DEF: Average velocity is the displacement covered per unit time.

(it obviously has direction,

the same as displacement)

SI unit : m/s

Instantaneous velocity is the velocity at one instant.

The speedometer of a car reveals information about the instantaneous speed of your car. It shows your speed at a particular instant in time.

If direction is included you have instantaneous velocity.

Average and Instantaneous Speed

How fast do your eyelids move when you blink? Displacement is zero, so vavg = 0. How fast do you drive in one hour if you drive zigzag and the magnitude of the displacement is different from distance?

To get the answers to these questions we introduce speed:

Speed is the distance object covers per unit time.

it tells us how fast the object is moving

KCR train has travelled a distance of about 6.7 km from University Station to Tai Po Market Station. But if we measure their separation by drawing a straight line on the map, we will find that Tai Po Market Station is only 5.4 km from University Station, roughly in the North-West direction. This is the displacement of the train.

We take the KCR trip from University Station to Tai Po Market Station. It takes about 6 minutes to travel a distance of 6.7 km. Thus,

The displacement of Tai Po Market Station from University Station is 5.4 km, so

in the North-West direction. This is smaller than the average speed of the train.

So why do we care of velocity at all? OK, it gives us direction what is very important (just imagine airplane controller with information only on speed of airplanes not on directions). But we saw that average speed is greater in general than magnitude of average velocity. So why is concept of velocity so important?

Because acceleration is a vector, all of equations are vector equations.

Acceleration can be in any direction to the velocity and the motion will depend on that.

ONLY:

if motion is 1-D without changing direction;

speed = magnitude of velocitybecause

distance traveled = magnitude of displacement

instantaneous speed = magnitude of instantaneous velocity

A racing car travels round a circular track of radius 100 m.

The car starts at O.

It travels from O to P in 20 s.

Its velocity was

Its speed was

10 m/s.

πr/t = 16 m/s.

The car starts at O. It travels from O back to O in 40 s.

Its velocity was

Its speed was

0 m/s.

2πr/t = 16 m/s.

Let’s look at the motion with constant velocity so called uniform motion

in that case, velocity is the same at all times so v = vavg at all times, therefore:

or x = vt

This is the only equation that we can use for the motion with constant velocity.

Object moving at constant velocity covers the same distance in the same interval of time.

Acceleration is the change in velocity per unit time.

(Change in velocity ÷ time taken)

vector quantity – direction of the change in velocity

In the SI system the unit is

meters per second per second.

a = 3 m/s2 means that velocity changes 3 m/s every second!!!!!!

If an object’s initial velocity is 4 m/s then after one second it will be 7 m/s, after two seconds 10 m/s, ….

Let’s look at the motion with constant acceleration so called uniformlly accelerated motion

let:

t = the time for which the body accelerates

a = acceleration

u = the velocity at time t = 0, the initial velocity

v = the velocity after time t, the final velocity

x = the displacement covered in time t

velocity v at any time t = initial velocity uincreased bya, every second

→

v = u + at

example:

u = 2 m/s

a = 3 m/s2

arithmetic sequence, so

speed increases 3 m/s EVERY second.

In general:

for the motion with constant acceleration:

Till now we had three formulas

From definition of average velocity we can find displacement in any case:

x = vavg t

For motion with constant acceleration, velocity changes according:

v = u + at

and average velocity is:

These three equations are enough to solve any problem in motion with constant acceleration.

But we are lazy and we want to have more equations that are nothing new, but only manipulations of this three.

Uniform Accelerated Motion – all together

1 – D Motion with Constant Acceleration

v = u + at

x = vavg t

for any motion

v2 = u2 + 2ax

In addition to these equations to solve a problem with constant acceleration you’ll need to introduce your own coordinate system, because displacement, velocity and acceleration are vectors (they have directions).

First 5 seconds of motion:

Slope of the graph =

Conclusion:

slope of the displacement – time graph is velocity

Therefore for the first 5 s: v = 50/5 =10 ms

vavg gives us no details of the motion between initial and final points.

Average velocity of a particle during the time interval Δt is equal to the slope of the straight line joining the initial (P) and final (T) position on the position-time graph.

First 5 seconds:

Slope of the graph =

Conclusion:

slope of the velocity – time graph is acceleration

Therefore for the first 5 s: a = 50/5 =10 ms²

Determine its displacement: a is constant,so

Conclusion: Area under a velocity-time graph is the displacement.

During the first 5 s the object has travelled: ½ x 50 x 5 = 125m

From displacement – time graph we can find velocity (instantaneous and average) by calculating the slope

The slope is the rate of change of acceleration (jerk).

Area under the graph v – t is the change in velocity.

The slope represent acceleration.

The area under graph represent its displacement.

Acceleration can cause: 1. speeding up 2. slowing down

3. and/or changing direction

So beware: both velocity and acceleration are vectors. Therefore

1. if velocity and acceleration (change in velocity) are in the same direction, speed of the body is increasing.

2. if velocity and acceleration (change in velocity) are in the opposite directions, speed of the body is decreasing.

3. If a car changes direction even at constant speed it is accelerating. Why? Because the direction of the car is changing and therefore its velocity is changing. If its velocity is changing then it must have acceleration.

This is sometimes difficult for people to grasp when they first meet the physics definition of acceleration because in everyday usage acceleration refers to something getting faster.

A stone is rotating around the center of a circle. The speed is constant, but velocity is not – direction is changing as the stone travels around, therefore it must have acceleration.

Velocity is tangential to the circular path at any time.

ACCELERATION IS ASSOCIATED WITH A FORCE!!!

The force (provided by the string) is forcing the stone to move in a circle giving it acceleration perpendicular to the motion – toward the CENTER OF THE CIRCLE - along the force. This is the acceleration that changes velocity by changing it direction only.

When the rope breaks, the stone goes off in the tangential straight-line path because no force acts on it.

In the case of moon acceleration is caused by gravitational force between the earth and the moon. So, acceleration is always toward the earth. That acceleration is changing velocity (direction only).

1. weakening gravitational force would result in the moon getting further and further away still circling around earth.

blue arrow –

velocity

red arrow – acceleration

2. no gravitational force all of a sudden: there wouldn’t be acceleration – therefore no changing the velocity (direction) of the moon, so moon flies away in the direction of the velocity at that position ( tangentially to the circle).

3. The moon has no speed – it moves toward the earth – accelerated motion in the straight line - crash

4. High speed – result the same as in the case of weakening gravitational force

Only the right speed and acceleration (gravitational force) would result in circular motion!!!!!!!

Free fall is vertical (up and/or down) motion of a body where gravitational force is the only or dominant force acting upon it.

(when air resistance can be ignored)

Gravitational force gives all bodies regardless of mass or shape, when air resistance can be ignored, the same acceleration.

This acceleration is called free fall or gravitational acceleration

(symbol g – due to gravity).

Free fall acceleration at Earth’s surface is about g = 9.8 m/s2 toward the center of the Earth.

Let’s throw an apple equipped with a speedometer upward with some initial speed.

That means that apple has velocity uas it leaves our hand.

The speed would decrease by 9.8 m/s every second on the way up, at the top it would reach zero, and increase by 9.8 m/s for each successive second on the way down

g depends on how far an object is from the center of the Earth.

The farther the object is, the weaker the attractive gravitational force is, and therefore the gravitational acceleration is smaller.

At the bottom of the valley you accelerate faster (very slightly) then on the top of the Himalayas.

Gravitational acceleration at the distance 330 km from the surface

of the Earth (where the space station is) is ̴7.8 m/s2.

In reality – good vacuum (a container with the air pumped out) can mimic this situation.

August 2, 1971 experiment was conducted on the Moon – David Scott – he simultaneously released geologist’s hammer and falcon’s feather. Falcon’s feather dropped like the hammer. They touched the surface at the same time.

1. Dr. Huff, a very strong lady, throws a ball upward with initial speed of 20 m/s.

How high will it go? How long will it take for the ball to come back?

Givens: Unknowns:

u = 20 m/s t = ?

g = - 10 m/s2 y = ?

at the top v = 0

2. Mr. Rutzen, hovering in a helicopter 200 m above our school suddenly drops his pen.

How much time will the students have to save themselves? What is the velocity/speed of the pen when it reaches the ground?

Givens:

u = 0 m/s (dropped)

g = 10 m/s2

Unknowns:

t = ?

v = ?

3. Mrs. Radja descending in a balloon at the speed of 5 m/s above our school drops her car keys from a height of 100 m.

How much time will the students have to save themselves?

What is the velocity of the keys when they reach the ground?

t = ?

v = ?

4. Dr. Huff, our very strong lady, goes to the roof and throws a ball upward. The ball leaves her hand with speed 20 m/s. Ignoring air resistance calculate

a. the time taken by the stone to reach its maximum height

b. the maximum height reached by the ball.

c. the height of the building is 60 m. How long does it take for the ball to reach the ground?

d. what is the speed of the ball as it reaches the ground?

d. v = u + gt

v = 20 – 10 x 6 = – 40 m/s

speed at the bottom is 40 m/s

Time Velocity Distance

(s) (m/s) (m)

+

0 0 0

1 10 5

2 20 20

3 30 45

4 40 80

u = 0 m/s

g = 10 m/s2

v= g t = 10t

= 5 t2

Velocity vs. time

Distance vs. time

40

80

30

60

velocity (m/s)

Distance (m)

40

20

10

20

0

0 1 2 3 4 5

0 1 2 3 4 5

Time (s)

Time (s)

constant slope → constant acceleration

changing slope – changing speed → acceleration

If air resistance can not be neglected, there is additional force (drag force) acting on the body in the direction opposite to velocity.

displacement

time

time

velocity

velocity

acceleration

acceleration

time

time

time

time

Comparison of free fall with no air resistance and with air resistance

In vacuum

In air

terminal velocity is maximum velocity an object can reach in air/any fluid.

Acceleration is getting smaller due to air resistance and eventually becomes zero.

When the force of the air resistance equals gravity, the object will stop accelerating and maintain the same speed.

It is different for different bodies.

Air Drag and Terminal Velocity

If a raindrops start in a cloud at a height h = 1200m above the surface of the earth they hit us at 340mi/h; serious damage would result if they did. Luckily: there is an air resistance preventing the raindrops from accelerating beyond certain speed called terminal speed….

How fast is a raindrop traveling when it hits the ground?

It travels at 7m/s (17 mi/h) after falling approximately only 6 m. This is a much “kinder and gentler” speed and is far less damaging than the 340mi/h calculated without drag.

The terminal speed for a skydiver is about 60 m/s (pretty terminal if you hit the deck)

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