Monday, February 7Monday, February 7

PHYS161PHYS161

Chapter 2Chapter 2

Quiz #2 TOPICSQuiz #2 TOPICS UncertaintiesUncertainties

Identifying appropriate uncertainty valuesIdentifying appropriate uncertainty values Carrying uncertainties through calculationsCarrying uncertainties through calculations

Motion diagramsMotion diagrams Converting verbal descriptions to motion diagrams and visa-Converting verbal descriptions to motion diagrams and visa-

versaversa Identifying directions of velocity and acceleration for various Identifying directions of velocity and acceleration for various

situationssituations Given two vectors, determine difference and sumGiven two vectors, determine difference and sum Meaning of “constant” accelerationMeaning of “constant” acceleration

GraphsGraphs position vs. velocity vs. accelerationposition vs. velocity vs. acceleration Equations for slope vs. definitions of velocity and accelerationEquations for slope vs. definitions of velocity and acceleration

UncertaintiesUncertainties

Identifying appropriate uncertainty valuesIdentifying appropriate uncertainty values Resolution (rounding)Resolution (rounding) Precision (repeatability)Precision (repeatability)

Carrying uncertainties through calculationsCarrying uncertainties through calculations For addition and subtraction, just add numerical For addition and subtraction, just add numerical

uncertaintiesuncertainties For multiplication or division, add relative For multiplication or division, add relative

uncertaintiesuncertainties convert numerical uncertainties to relative uncertaintiesconvert numerical uncertainties to relative uncertainties add relative uncertaintiesadd relative uncertainties convert total back to numerical uncertaintyconvert total back to numerical uncertainty

ExamplesExamples

What is the likely uncertainty associated What is the likely uncertainty associated with the resolution of the following with the resolution of the following measurements?measurements?A device provides you with a measurement A device provides you with a measurement

value of 2.54 g.value of 2.54 g.A ruler with marks every millimeter.A ruler with marks every millimeter.

ExamplesExamples

You have three measurements:You have three measurements:aa = (1.0 m = (1.0 m 0.01 m) 0.01 m)

bb = (2.0 m = (2.0 m 0.01 m) 0.01 m)

cc = (2.0 m = (2.0 m 0.02 m) 0.02 m)

What is What is aa++bb--cc??

What is What is abcabc??

What is What is bb22/c/c??

What is What is √√(bc)?(bc)?

Motion diagramsMotion diagrams

Location at regular time intervals is Location at regular time intervals is represented by dots.represented by dots.

The average velocity during each time The average velocity during each time interval is represented by an arrow interval is represented by an arrow between the two dotsbetween the two dots

The average acceleration for the double The average acceleration for the double time interval centered on each dot is time interval centered on each dot is represented by an arrow coincident with represented by an arrow coincident with each doteach dot

DirectionsDirections

The direction of the velocity is in the direction of The direction of the velocity is in the direction of motion.motion.

The direction of the acceleration is in the direction The direction of the acceleration is in the direction of the of the change change in velocityin velocity..Acceleration is same direction as velocity if object is Acceleration is same direction as velocity if object is

speeding upspeeding upAcceleration is opposite direction of velocity if object is Acceleration is opposite direction of velocity if object is

slowing downslowing downAcceleration is perpendicular to direction of velocity if Acceleration is perpendicular to direction of velocity if

object is changing directions (and maintaining the object is changing directions (and maintaining the same speed)same speed)

ExamplesExamples

A rock is thrown up in the air. On the way up, it A rock is thrown up in the air. On the way up, it slows down. What is the direction of the rock’s slows down. What is the direction of the rock’s acceleration as it slows down?acceleration as it slows down?

When the cart rolls down the incline, is the When the cart rolls down the incline, is the acceleration of the cart constant? Is the velocity acceleration of the cart constant? Is the velocity of the cart constant?of the cart constant?

Is it possible for an object’s acceleration to be Is it possible for an object’s acceleration to be constant in direction and magnitude yet the constant in direction and magnitude yet the velocity decrease?velocity decrease?

Is it possible for an object’s acceleration to be Is it possible for an object’s acceleration to be constant in direction and constant in direction and decreasing decreasing in in magnitude yet the velocity magnitude yet the velocity increaseincrease??

DefinitionsDefinitions

Velocity = the rate at which the position Velocity = the rate at which the position changeschanges

v v = = dsds//dtdt

Average velocity = the average rate at which Average velocity = the average rate at which the position changes during a time intervalthe position changes during a time interval

vvavgavg = = ΔΔss//ΔΔtt

DefinitionsDefinitions

Acceleration = the rate at which the velocity Acceleration = the rate at which the velocity changeschanges

a a = = dvdv//dtdt

Average acceleration = the average rate at Average acceleration = the average rate at which the velocity changes during a time which the velocity changes during a time intervalinterval

aaavgavg = = ΔΔvv//ΔΔtt

ExamplesExamples

At a particular time, a car is traveling north at a At a particular time, a car is traveling north at a speed of 20 mph. An hour later, the car is 10 speed of 20 mph. An hour later, the car is 10 miles east of the starting point, traveling south miles east of the starting point, traveling south at a speed of 20 mph.at a speed of 20 mph.

1.1. What is the instantaneous velocity of the car at What is the instantaneous velocity of the car at the initial time?the initial time?

2.2. What is the average velocity of the car during What is the average velocity of the car during the hour?the hour?

3.3. What is the average acceleration of the car What is the average acceleration of the car during the hour?during the hour?

ExampleExample

Drop a ball such that the time is recorded in three places Drop a ball such that the time is recorded in three places ((tt11, , tt22 and and tt33):):

1.1. What is the average velocity of the ball during the time What is the average velocity of the ball during the time interval from interval from tt11 to to tt22??

2.2. What is the average velocity of the ball during the time What is the average velocity of the ball during the time interval from interval from tt11 to to tt33??

3.3. Assuming the acceleration is constant, at what time Assuming the acceleration is constant, at what time does the ball have an instantaneous velocity equal to does the ball have an instantaneous velocity equal to the value obtained in #1? How about an instantaneous the value obtained in #1? How about an instantaneous velocity equal to the value obtained in #2?velocity equal to the value obtained in #2?

4.4. Assuming the acceleration is constant, what is the Assuming the acceleration is constant, what is the acceleration of the ball as it fell?acceleration of the ball as it fell?

GraphsGraphs

Given a graph of a straight line, determine the Given a graph of a straight line, determine the equation that represents the relationship equation that represents the relationship between the two variablesbetween the two variables

Given position vs. time graph, determine velocity Given position vs. time graph, determine velocity graph. Similarly, given velocity vs. time graph, graph. Similarly, given velocity vs. time graph, determine acceleration graph.determine acceleration graph.

Determine what shape of a position vs. time Determine what shape of a position vs. time graph is consistent with a particular velocity vs. graph is consistent with a particular velocity vs. time graph. Similarly, given acceleration vs. time time graph. Similarly, given acceleration vs. time graph, determine shape of velocity graph.graph, determine shape of velocity graph.

SlopeSlope

Slope Slope (from one point to the another point) = (from one point to the another point) = change in vertical value divided by change change in vertical value divided by change in horizontal valuein horizontal value

Using Using y y for the vertical value and for the vertical value and xx for the for the horizontal value, horizontal value, slope slope (from one point to (from one point to the another point) the another point) = = ΔΔyy//ΔΔxx

Slope for position vs. time graphsSlope for position vs. time graphs

Using Using y y for the vertical value and for the vertical value and xx for the for the horizontal value, horizontal value, slope slope (from one point to (from one point to the another point) the another point) = = ΔΔyy//ΔΔxx

If we plot position (If we plot position (ss) as the vertical value ) as the vertical value and time (and time (tt) as the horizontal value then ) as the horizontal value then slope slope (during the time interval (during the time interval ΔΔtt) ) = = ΔΔss//ΔΔtt

Slope and velocitySlope and velocity

If we plot position (If we plot position (ss) as the vertical value ) as the vertical value and time (and time (tt) as the horizontal value then ) as the horizontal value then slope slope (during the time interval (during the time interval ΔΔtt) ) = = ΔΔss//ΔΔtt

If we define average velocity as the rate at If we define average velocity as the rate at which the position changes during a time which the position changes during a time interval (interval (ΔΔss//ΔΔtt) then the ) then the slope of a position slope of a position vs. time graph vs. time graph (during the time interval (during the time interval ΔΔtt) ) == vvavg avg (during the time interval)(during the time interval)

Slope at a pointSlope at a point

Slope Slope (at a single point) = infinitesimal (at a single point) = infinitesimal change in vertical value at that point change in vertical value at that point divided by infinitesimal change in divided by infinitesimal change in horizontal value at that point horizontal value at that point = = ddyy/d/dxx

For position vs. time graph, the For position vs. time graph, the slope slope (at an (at an instant) instant) = d= dss/d/dt = vt = v (at that instant) (at that instant)