3. KinematicsRevF.pptx

Course Coverage and Status



√1. Introduction



√2. Physical Quantities and Vectors



3. Kinematics



4. Forces & Universal Gravitation



5. Work, Energy and Power



6. Linear Momentum and Collision



7. Fluid Mechanics

Chapter OUTLINE

Kinematics

Kinematics

• Brief History • Definition

1- Dimensional Kinematics

1- Dimensional Kinematics

• Distance & Speed

• Displacement, Velocity & Acceleration

2-Dimensional Kinematics

2-Dimensional Kinematics

• Projectile Motion

Chapter

Objectives

describe the theory of Mechanics

(Newtonian) and its branches -

Kinematics and Dynamics

describe the theory of Mechanics

(Newtonian) and its branches -

Kinematics and Dynamics

define the following concepts and

differentiate them from one another:

distance, displacement, speed,

velocity & acceleration

define the following concepts and

differentiate them from one another:

distance, displacement, speed,

velocity & acceleration

apply this concepts in solving kinematic

problems

Brief Historical Development

Aristotle

Aristotle Galileo Galilei



Aristotle

 “There are only two types of motion, the natural motion and the violent motion”

 Natural- proceed from the “nature of objects”

“objects were thought to fall at speeds proportional to their weights”

 Violent – resulted from pushing or pulling

“all motions resulted either from the nature of the moving object or from a sustained push or pull”

 Galileo Galilee

Better Observation: Galileo

 Studied motion down a ramp: imagined a rolling ball without friction: would continue indefinitely, without being pushed!

 In particular, Galileo

compared the times for the

full distance roll and that for

one-quarter of the full distance.

Galileo found that in twice the time, the ball rolled four times the

Newton Extended Galileo’s Picture of

Motion to Include Forces

Galileo:

 Natural horizontal motion is at constant velocity unless a force acts: a push from behind will cause acceleration, friction will cause negative acceleration (that is,

deceleration).

 Natural vertical motion is constant downward

Newton Said

 The “natural vertical motion” is at constant acceleration because there’s a constant force acting – the force of gravity!

 Without that force, vertical motion would be at constant velocity.

 Look again at the path of a projectile: without gravity, it would be a straight line.

 _{Newton established}

3

MECHANICS

The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts.

KINEMATICS

deals with the description of motion.

DYNAMICS



Motion along a straight line



the simplest case of motion to study



considering objects to travel along a flat and

narrow straight line



Object in consideration is termed as

point

particle

REFERENCE FRAME



Reference frame

– a physical entity to

which the motion and position of objects

are referred.



Inertial frame

– a reference frame moving

Inertial Frames of Reference

 Recall a frame of reference is a set of axes, like three perpendicular rulers, to measure

position, plus a clock to track time, so motion can be precisely described.

 An inertial frame is one in which Newton’s

First Law is obeyed.

COORDINATE SYSTEM

- a system for

assigning numbers or

coordinates to the

location of a point in a reference frame.

 Position is defined in

terms of a frame of reference



One

KINEMATIC CONCEPTS

DISTANCE

(x)

DISPLACEMENT

( x = x

∆

- x

)

-a SCALAR quantity -a VECTOR quantity

• _{"how much space an object}

has covered?

•_{"how far out of place an object}

is?"

total length traveled change in position

shortest distance from initial

position to final position of the object being considered

Displacement



Displacement

measures the

change in

position



Represented as



x (if horizontal) or



y (if vertical)



Vector quantity



+ or - is generally sufficient to

indicate direction for

one-dimensional motion

Note:



Distance may be, but is not necessarily, the

magnitude of the displacement

Distance

(blue line)

Displacement

Position-time graphs

Consider this



If you run a 1000m circular race track

from start to finish,



What is your total distance

travelled?

 1000m



What is your total displacement?

KINEMATIC CONCEPTS

- a SCALAR quantity

- " how fast an object is moving?”

SPEED

VELOCITY

- a VECTOR quantity

- " the rate at which an object moves changes its

position”

_Units:

m/s,

V = total displacement/ total time

t

x

x

v





 Average Speed

 Distance traveled

along the path of an object divided by the time it takes to travel the distance.

 Average Velocity

 Displacement of an

Example:

Suppose that in both cases truck covers the distance in 10 seconds. What is the truck’s velocity for each trip?

EXAMPLE:

 Mario enters in a kart race. He completes the 1250m race around a circular track in

31.7532 s

 1. What is Mario’s average speed?

 2. What is Mario’s average velocity?



You run 100m in 12 s, then turn

around and jog 50m back toward the

starting point in 30s. Calculate



(a) your average speed, and



(b) your average velocity for the total

trip.



t



d

* Concern only on the initial point and final point Graphical Representation:

Average Speed or Average Velocity

t

d

v

Graphical Interpretation of Average Velocity

 Velocity can be determined from a position-time

graph

 Average velocity equals the slope of the line

Instantaneous Velocity

 Instantaneous velocity is defined as the limit

of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero

 The instantaneous velocity indicates what is

happening at every point of time (or at a given instant)

t

x

x

t

x

v

Graphical Interpretation of

Instantaneous Velocity

 Instantaneous velocity is the slope of the

tangent line to the curve at the time of interest

 The magnitude of the instantaneous velocity is

Average vs Instantaneous

Velocity

Uniform Velocity

 Uniform velocity is constant

velocity  The

instantaneous velocities are

always the same

 All the

Acceleration

-

a

VECTOR

quantity

-

“the rate at which

an object

changes

its velocity”

-

dimension: L/T

t

v

a







Note:



Acceleration has

nothing to do with

moving fast!!!!



If an object is slowing down, then its

acceleration is in the opposite direction of

its motion.



For motion with

constant velocity

, the

Falling Apples



A falling apple captured by

strobe photography at 60

flashes per second.



The acceleration of the

Average Acceleration



Changing velocity (non-uniform) means an

acceleration is present



Average acceleration

is the

rate of change

of the velocity



Average acceleration is a

vector

quantity (i.e.

described by both magnitude and direction)

t

v

v

t

v

a

Average Acceleration



When the sign of the velocity and the

acceleration

are the same

(either positive

or negative), then the

speed is increasing

When the sign of the velocity and the

acceleration are

opposite

, the

speed is

decreasing

Units

SI Meters per second squared (m/s2)

CGS Centimeters per second squared (cm/s2)

Instantaneous and Uniform

Acceleration



Instantaneous acceleration

is the limit

of the average acceleration as the time

interval goes to zero

 The instantaneous accelerations will all be

equal to the average acceleration

0 0

lim

lim

inst _{t t}

Graphical Interpretation of Acceleration

 Average acceleration is

the slope of the line

connecting the initial and final velocities on a

velocity-time graph

 Instantaneous

Instantaneous Acceleration



This is just like the definition of instantaneous

velocity:



The acceleration at time

t

is the

slope of the velocity graph

v

(

t

)

at that time.

t t₁

O

Describing motion with

motion diagrams and graphs

Constant velocity – velocity (magnitude and direction) is not changing with time

Constant acceleration – acceleration (magnitude and direction) is not changing with time (Also termed as uniformly accelerated motion (UAM)

Zero

accelerationZero

acceleration

Non - zero acceleration

Constant Acceleration

 Constant acceleration means the rate of change of velocity is constant.

 The solution to this equation is

 Example:: A car traveling at 10 m/s accelerates

steadily at 2 m/s2. How fast is it going after 2 sec?

After 4 sec?

 V = Vo + at = 10m/s + 2 m/s2 (2s) = 14m/s (after 2 secs)

 V = 10m/s + 2 m/s2 (4s) = 18m/s (after 4 sec)

constant.

dv

a

dt

 

0

.

Distance Moved at Constant

Acceleration

 At constant acceleration,

 The solution of this equation is



Here

x

is the beginning position,

v

the

beginning velocity,

a

the constant

acceleration.

0

( )

.

dx

v t

v

at

dt



 

2 1

0 0 2

( )

.

More about Constant Acceleration…

 At constant acceleration, the graph of velocity as a

function of time v(t) = v₀ + at

is a straight line:

 If v = v₀ at t = 0, and v = v₁ at t = t₁, the average

velocity over the time interval 0 to t₁ is

 IMPORTANT! This formula is unlikely to be correct at

nonconstant acceleration.

0

v₀ v(t)

t₁ v₁

0 1

.

2

v

v

Constant acceleration

Y-intercept = Vo

Note that V = Vo + at

Constant Acceleration

Formulas

These formulas are worth memorizing: the last one is simply derived by eliminating t between the first two.

0

v v

 

at

2 1

0 0 2

x x v t

  

at

0 1

2

v

v

v









2 2

0

2

Example 1: Motion Diagrams



Uniform velocity

(shown by red arrows

maintaining the same size)

Example 2:

 Velocity and acceleration are in the same direction

 Velocity is increasing (red arrows are getting longer)

 Acceleration is uniform (blue arrows maintain the

Example 3:

 Acceleration and velocity are in opposite directions  Velocity is decreasing (red arrows are getting shorter)  Acceleration is uniform (blue arrows maintain the

Examples:GRAPHS

time position

time velocity

Example: position vs time

graphs

t d

t d

t d

Example: velocity vs time

graphs

t v

t v

t v

What is the average acceleration…

 during the first 10 minutes? = 60-0/10 = 6m/min²

 between 10 and 15 minutes? = 60-60/5 = 0

 between 15 to 40 minutes? = -40 – 60/25 = - 4m/min²