Intro to Physics1D Kinematics

Pre AP Physics

International System (SI) of Units The number is only part of the answer; it

also needs UNITS In 1960, the International System of units

(or SI) was adopted as the standard There are seven SI base units from which all

others can be derived. The seven base (fundamental) units

describe, in numbers, the seven basic quantities (dimensions).

Pre AP Physics

SI Base Units• Laws of physics are expressed in basic

quantities that require a precise definition• In mechanics, the 3 basic quantities are:• length (meter) – the distance traveled by

light in a vacuum during 1/ 299,792,458 s. • time (second) – the time it takes cesium-133

to vibrate 9,192,631,770 times• mass (kilogram) – kilogram (kg) defined as

the mass of a specific (very stable) Pt-Ir alloy cylinder

Pre AP Physics

Four Additional SI Units• current (ampere) – one coulomb of charge flowing

through a wire in one second• amount of substance (mole) – 6.02 x 1023 things• temperature (Kelvin) – 1/273.16 of the

temperature of the triple point of water (where solid, liquid, and gaseous forms can exist at the same time

• luminosity (candela) – luminous intensity of 1/60 cm2 of black body radiation operating at the temperature of solidification of platinum

Pre AP Physics

Prefixes

• Labels attached to the beginning of a unit name, multiply the unit by the associated conversion factor.

• Prefixes designate a multiple of the base unit in powers of ten.

• One power of ten is one order of magnitude– (You should be able to use prefixes from Giga

to pico)

Pre AP Physics

Kinematics I - Velocity

Position and Distance

Pre AP Physics

Kinematics• Is the study of motion without regard for the

forces causing the motion or …• the description of motion using:

Position (s, x, y, r)– The location of an object in space– changes in position can be described by distance or

displacement

Velocity (v)– how fast an object is changing its position

Acceleration (a)– how fast the velocity of an object is changing

Pre AP Physics

POSITION

• Must be defined relative to a reference point.• Must be defined using both magnitude and

direction. • Position (s, x, y, r) is the location of an object in

space• units: m, cm, km

Pre AP Physics

Distance

• Distance is the separation between any two points.• It is a scalar quantity, defined only by magnitude.

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What is a Scalar?

• A scalar is a quantity that is described only by magnitude…such as distance. – it may have units e.g. kg (it may not)– it does not have any direction– it may have a sign + or –– it may be real, or integer

– The distance from All Saints to Noonday is about 10 kilometers (as the crow flies).

Pre AP Physics

Displacement • A objects displacement is defined as the separation

between an object and a reference point.• It is a vector quantity, defined by magnitude and

direction. • Displacement is the change in position of an

object (∆s = sf - si)

si

sf

∆d = sf – si

displacement = ∆s∆s

Pre AP Physics

What Is a Vector?• A vector is a quantity that must be described by

magnitude and direction.• Magnitude

– Size or quantity in unit of measurement– 10 meters, 5 Newtons

• Direction– Compass direction: North, Southeast– Angle: 25° from the positive x-axis

Pre AP Physics

Vector– an arrow drawn to scale used to represent a

vector quantity

–vector notation

Pre AP Physics

Graphical (Tip-to-Tail ) Method

• Example 1: Add these vectors using the tip-to-tail method.

+

Pre AP Physics

Graphical (Tip-to-Tail) Method

+ =

Is the resultant

Pre AP Physics

What is a Resultant?

Pre AP Physics

Vector Addition in One Dimension

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How does one solve for displacement?∆s = sf – si∆s = (x1, y1) – (x2, y2) How do you do this?

• Realize that displacement is a vector so you must decide to use either the Cartesian or polar coordinates

∆s

si = (xi, yi)

sf = (xf, yf)

∆s = sf – si

X

Y

Pre AP Physics

Describing Vectors

• Cartesian Coordinates (rx, ry)

• rx = x2 – x1 = distance in the x-direction

• ry = y2 – y1 = distance in the y-direction

• Polar Coordinates (r, θ)

• “How far and in which direction”

θ = measured directly from graphPre AP Physics

r =p(x2 � x1)2 + (y2 � y1)2

Describing Distance

leg 1 = 2 miles

leg 3 = 2 miles

leg 2 = 3 miles

N

Total DISTANCE Traveled= 2 miles + 3 miles + 2 miles

= 7 milesPre AP Physics

Describing Displacement

N

displa

cemen

t vec

tor

Describing Displacement

First Method (Cartesian) 3 miles East 4 miles North (3, 4) milesput ‘horizontal’ coordinate 1st

put ‘vertical’ coordinate 2nd

Pre AP Physics

Displacement Magnitude

N

displa

cemen

t vec

tor

3 miles

4 m

iles

Second Method (Polar)1st - calculate length ofdisplacement vector

Pre AP Physics

~r =p

32 + 42

~r =p25

~r = 5 miles

Displacement Direction

N

displa

cemen

t vec

tor

θ3 miles

4 m

iles

2nd – Calculate the angle using trigonometric relationships

Pre AP Physics

✓ = tan

�1

✓opp

adj

◆

✓ = tan�1

✓4 miles

3 miles

◆= 53.1�

Displacement Vector(Polar Notation)

Describe the displacement vector by its length and direction

N

displa

cemen

t vec

tor

θ3 miles

4 m

iles

Pre AP Physics

~r = 5 miles, 53.1�

Kinematics I - Velocity

Graphing

Pre AP Physics

Objectives for Graphing

• Measure, graph, analyze, and interpret laboratory data.

• Identify the relationships that may exist between variables (direct, inverse, and quadratic) and the graph shapes, which indicate these relationships (straight line, hyperbola, and parabola).

Pre AP Physics

GRAPHING • Graphing is the pictorial relationship between

quantities

– coordinate = any point w/in a graph.– Slope = rise / run or ∆ y / ∆ x ∆ = means change x = horizontal axis y = vertical axis

Pre AP Physics

All relationships involve only two variables.

• The Independent Variable – – this is the variable that you control or manipulate. – It is almost always plotted on the x-axis.

• The Dependent Variable – – this is the variable that responds to the independent

variable. – It is usually plotted in the y-axis.

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Graphical Analysis

• Graphical analysis provides a tangible representation of a relationship from which you can make predictions and inferences.– Mass of a substance vs. volume– Speed of a car vs. braking distance– Amount of light vs. plant growth

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What are the rules?

• The graph has a title…• The scale of the graph is determined by the range

of the data and does not need to start at 0,0• The independent variable is plotted on the x-axis• The dependent variable is plotted on the y-axis• All variables have names and units• You construct a best fit line through the points

Pre AP Physics

Linear Relationships

• The dependent variable varies directly with the independent variable.

• Plot is a straight line.• y ∝ x or y = kx• y = mx + b

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Parabolic (Exponential) Relationship

• The dependent variable varies with the square of the independent variable.

• y ∝ x2 or y = kx2

• Plot does not appear as a straight line.

• Note that one variable changes faster than the other resulting in a curved line.

Pre AP Physics

Inverse (Hyperbolic) Relationships

• The dependent variable varies in the opposite direction of the independent variable.

• As x goes up, y goes down.• y ∝ 1 ⁄ x or y = k ⁄ x• This results in a line that

starts high and ends low. It can be straight or curved.

Pre AP Physics

Graphical Analysis

Pre AP Physics

Slope expresses the relationship between the variables.

• How y is changing as x is manipulated.

• m = ∆y ⁄ ∆x• ∆y = y2 – y1

• ∆x = x2 – x1

• The units of the slope define the meaning of the slope.

Pre AP Physics

Area Under the Curve

• The shaded area is the area under the curve.

• You determine the area by adding up the areas of each triangle and square.

• Again, the units of this calculation, define what the number represents.

Pre AP Physics

Interpolation and Extrapolation

• Interpolation is the reading or analysis (inference or prediction) of the graph from within the plotted data points.

• Extrapolation is the reading or analysis (inference or prediction) of the graph from outside of the plotted data points. This can lead to a significant source of error.

Pre AP Physics

Kinematics I - Velocity

Position-Time Graphs

Pre AP Physics

Graphing the Relationships Between s, v, & a• s is the position • v is the rate of change of s with time• a is the rate of change of v with time• consider a graph of s vs. time

– s on vertical axis– time on horizontal axis– rate of change is interpreted as the slope

Pre AP Physics

Position vs. Time for Constant Velocity

Pre AP Physics

Position vs. Time for Constant Velocity

Pre AP Physics

Position vs. Time for Acceleration

Pre AP Physics

Slope of Position vs. Time is Velocity

Pre AP Physics

Changes in the slope

• quick change– very steep slope

•positive slope–up and to the right

•negative slope–down and to the right

•slow change–very flat slope

Pre AP Physics

Slope Practice 1

• Calculate the velocity of the object whose graph is shown above.

• –24 m / 8 s = –3 m/s

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Slope Practice 2

• Determine the velocity of the object whose motion is graphed at the left.

• 20 m / 5 s= 4 m/s

Pre AP Physics

Kinematics I - Velocity

SpeedVs

Velocity

Pre AP Physics

Speed• A scalar quantity referring to how fast an

object is moving. – A fast-moving object has a high speed while a

slow-moving object has a low speed. – An object with no movement at all has a zero

speed. • The distance traveled per unit time. • Its units are meters per second (m/s).

Pre AP Physics

v =s

t

Velocity

• Velocity is a vector quantity describing displacement per unit time.

• Its units are meters per second (m/s).• It is defined relative to a reference point.• Velocity describes both speed and direction.• Depending on your direction, it can be

negative or positive.

Pre AP Physics

Average Speed

• Total distance divided by total time of travel; • Rate you would have to travel constantly to cover

the same distance in the same time

Pre AP Physics

v̄ =s

t

Average Velocity• Total displacement divided by total time of travel;• Rate you would have to travel constantly to cover

the same displacement in the same time

Pre AP Physics

v̄ =�~s

�t=

~sf � ~sitf � ti

Kinematics I - Velocity

Instantaneous Velocity

Pre AP Physics

Average and Instantaneous• instantaneous speed – speed at any given instant

in time; speedometer reading, for example• instantaneous velocity - specifies how fast and in

what direction one is moving at one particular point in time

• magnitude of instantaneous velocity is exactly the same as instantaneous speed

Pre AP Physics

From a Graph• Instantaneous Velocity

– the slope of the displacement versus time graph– Determine the average velocity– Take average between smaller

and smaller intervals– Until you get to the

tangent line– The slope of the tangent

line is the instantaneous velocity at that point in time

Pre AP Physics

Kinematics I - Velocity

Problem Solution

Pre AP Physics

Problem Solution Guidelines• Draw a sketch

– Indicate origin and positive direction• List the given quantities using the symbols of the

equations. (si, vi, a)• Identify the unknown• Write the general equation of kinematics you will

be using, i.e.

Pre AP Physics

vf = vi + at s = si + vit+12at2

More Guidelines

• Isolate the unknown.• Rewrite the isolated equation using the known

quantities.• Check your units• Make sure you are answering the question asked.

– Your answer may be perfectly correct for the wrong question.

Pre AP Physics

Kinematics I - Velocity

Velocity-Time Graphs

Pre AP Physics

Velocity – Time for Constant Velocity

Pre AP Physics

Velocity – Time for Constant Acceleration

Time (s)

Velocity(m/s)

0 01 102 203 304 405 50

Pre AP Physics

Velocity – Time for Zero Acceleration

Time (s)

Velocity(m/s)

0 101 102 103 104 105 10

Pre AP Physics

Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 6 seconds.

Pre AP Physics

Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 4 seconds.

Pre AP Physics

Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 2 seconds to 5 seconds.

Pre AP Physics

Constant Positive Velocity

Pre AP Physics

Constant Negative Velocity

Pre AP Physics

Positive Velocity Positive Acceleration

Pre AP Physics

Positive Velocity Negative Acceleration

Pre AP Physics

Negative Velocity Negative Acceleration

Pre AP Physics

Negative Velocity Positive Acceleration

Pre AP Physics

Speeding Up & Slowing Down

Negative acceleration can mean speeding up or slowing down. The same is true with positive acceleration.

Pre AP Physics

Passing Lane – Position-Time

Pre AP Physics

Passing Lane – Velocity-Time

Pre AP Physics

The End!

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