Ch 5 – A Mathematical Ch 5 – A Mathematical Model of MotionModel of Motion

Graphing Motion in One DimensionGraphing Motion in One DimensionGraphing Velocity in One DimensionGraphing Velocity in One Dimension

AccelerationAccelerationFree FallFree Fall

5.1 Graphing Motion in One 5.1 Graphing Motion in One DimensionDimension

Position-Time GraphsPosition-Time Graphs Representing motion with Representing motion with

graphs…graphs… Pay attention to axes and Pay attention to axes and

units!units! Always ask yourself 2 Q’s: Always ask yourself 2 Q’s:

What is the object’s motion? What is the object’s motion? What does the slope What does the slope represent?represent? Slope = rise / run, or m/s for Slope = rise / run, or m/s for

d-t graphs, which is d-t graphs, which is velocity!velocity! Uniform (constant) motion Uniform (constant) motion

can be identified by an can be identified by an unchanging slopeunchanging slope..

5.1 Graphing Motion in One 5.1 Graphing Motion in One Dimension (cont.)Dimension (cont.)

Using an Equation to Find Out Using an Equation to Find Out WhereWhere and and WhenWhen Motion can also be represented by an algebraic Motion can also be represented by an algebraic

equation:equation:

vv = = dd / / t (where t (where dd = d = d11 – d – doo))

Recall that this is the equation for average velocity. Recall that this is the equation for average velocity. You can rearrange this equation to find the position of You can rearrange this equation to find the position of an object with an object with uniform motionuniform motion::

dd11 = d = doo + vt or d = d + vt or d = doo + vt + vt

You now have two equations of motion that will allow You now have two equations of motion that will allow you to solve you to solve wherewhere and and whenwhen for describing the for describing the motion of an object motion of an object traveling uniformlytraveling uniformly..

5.2 Graphing Velocity in One 5.2 Graphing Velocity in One Dimension Dimension

Instantaneous VelocityInstantaneous Velocity What if the motion is not What if the motion is not

uniform? (Changing)uniform? (Changing) The slope is changing so The slope is changing so

the velocity must be as well the velocity must be as well – – accelerationacceleration..

Ask again: What is the Ask again: What is the object doing? What does object doing? What does the slope represent? the slope represent? Must use a new technique Must use a new technique

for slope, but this time it is for slope, but this time it is not constant; you are not constant; you are finding a velocity at a given finding a velocity at a given time – time – instantaneous instantaneous velocityvelocity!!

5.2 Graphing Velocity in One 5.2 Graphing Velocity in One DimensionDimension

Velocity–Time Graphs Velocity–Time Graphs Same two questions: What is Same two questions: What is

the object doing? What does the object doing? What does the slope represent?the slope represent?

Additionally, you can also Additionally, you can also find find displacementdisplacement on a V-T on a V-T graph using a graph using a newnew technique: technique: area under the curve.area under the curve. The displacement for a given The displacement for a given

time interval is the time interval is the area area under the curveunder the curve of a V-T of a V-T graph. (pg.743) graph. (pg.743)

This leads to a 3This leads to a 3rdrd Q: What Q: What does the area under the does the area under the curve represent? curve represent? DisplacementDisplacement

Chapter 5 Assignment:Chapter 5 Assignment:

AC+P 1 pp. 108-111AC+P 1 pp. 108-111 AC # 16,18,19,21AC # 16,18,19,21 P’s # 27,28,29*,31,34,40P’s # 27,28,29*,31,34,40

* Use Logger Pro for this problem.* Use Logger Pro for this problem.

5.3 Acceleration5.3 Acceleration Determining Average Determining Average

AccelerationAcceleration Average acceleration is the rate Average acceleration is the rate

of change of velocity during a of change of velocity during a time interval.time interval. a = a = v v / / t (from chapter 3) t (from chapter 3)

Example: A car starts from rest Example: A car starts from rest and after 3 seconds is traveling and after 3 seconds is traveling 60 mph. BMW? Porsche? Audi? 60 mph. BMW? Porsche? Audi? What is the average acceleration What is the average acceleration in mph/sec? in mph/sec? v = v = vv11 – v – v0 0 = 60 mph and = 60 mph and

t = tt = t11 – t – t00 = 3 seconds = 3 secondsa = a = v / v / t = 60 mph / 3 sec = t = 60 mph / 3 sec =

20 mph / sec20 mph / sec

5.3 Acceleration (cont.)5.3 Acceleration (cont.)

Constant and Instantaneous Constant and Instantaneous Acceleration (again)Acceleration (again)

Ask again: what does the Ask again: what does the slope represent? What does slope represent? What does the area under the curve the area under the curve represent?represent?

Instantaneous acceleration Instantaneous acceleration for a non-constant graph for a non-constant graph would be done by calculating would be done by calculating slope of the tangent! (ex. slope of the tangent! (ex. page 95)page 95)

Pay attention to the axes!Pay attention to the axes!

5.3 Acceleration (cont.)5.3 Acceleration (cont.)

Positive and Negative AccelerationPositive and Negative Acceleration Pay attention to the chosen coordinate system when Pay attention to the chosen coordinate system when

working with diagrams and graphs. With equations working with diagrams and graphs. With equations the sign will take care of itself. (Fig 5-13)the sign will take care of itself. (Fig 5-13)

Equations and Examples for Objects with Equations and Examples for Objects with Constant Constant AccelerationAcceleration Velocity: v = vVelocity: v = v00 + at + at DisplacementDisplacement

d = dd = d00 + ½ (v + v + ½ (v + v00)t )t d = dd = d00 + v + v00t + ½ att + ½ at22

Final velocity: vFinal velocity: v22 = v = v0022 + 2a (d – d + 2a (d – d00))

Practice Problem #27 pg. 103

5.4 Free Fall5.4 Free Fall Acceleration Due to GravityAcceleration Due to Gravity

Galileo GalileiGalileo Galilei Recognized that motion of Recognized that motion of

falling objects could only be falling objects could only be understood if the effects of understood if the effects of air, water, or whatever air, water, or whatever medium the object was falling medium the object was falling through, were ignored.through, were ignored.

a = g = - 9.8 m/sa = g = - 9.8 m/s22 if upward is if upward is defined as a positive position defined as a positive position change and downward as the change and downward as the negative.negative.

For equations of accelerating For equations of accelerating objects in free fall, simply objects in free fall, simply substitute ‘a’ with ‘g’ and use substitute ‘a’ with ‘g’ and use as you would with other as you would with other physics problems.physics problems.

e.g v = ve.g v = v0 0 + gt+ gt

Practice Problem #31 pg. 106

Chapter 5 Assignments:Chapter 5 Assignments:

AC+P 1 pp. 108-111AC+P 1 pp. 108-111 AC # 16,18,19,21AC # 16,18,19,21 P’s # 27,28,29*,31,34,40,41,42P’s # 27,28,29*,31,34,40,41,42

AC+P 2 pp. 109-114AC+P 2 pp. 109-114 AC # 22, 26AC # 22, 26 P’s # 44, 45, 46, 47, 51, 57, 60*, 63a,b*, 67,69, P’s # 44, 45, 46, 47, 51, 57, 60*, 63a,b*, 67,69,

72* 72*

* = Use Logger Pro for these problems* = Use Logger Pro for these problems