Acceleration and Graphical Interpretations Related To Motion

1
Acceleration and Graphical Interpretations
Related To Motion

• Week 2 -- Lesson 5

2
Objectives
By the end of this set of notes, you should be
able to

• Define what is meant by the term Acceleration
• Discuss the difference between an average and an
  instantaneous value as well as determine these
  types of values graphically
• Discuss the motion of an object given a plot of
  its position or velocity as a function of time

3
Objectives
By the end of this set of notes and lab
experience you should be able to

• Sketch a graph of an objects position or
  velocity as a function of time given information
  about the motion of that object.

4
Acceleration

• Definition The acceleration of an object is the
  time rate of change of velocity of that object.
• Acceleration is a vector quantity.
• Unit m/s2 Dimension L/T2

5
The Unit Of Acceleration

• The unit of acceleration is read meter per
  second per second rather than meter per second
  squared.
• I have no idea what a second squared nor square
  second is. However, if you read 3 m/s2 as 3
  (meters per second) per second, the meaning is a
  bit more apparent the speed of the object is
  changing by 3 meters per second in each second of
  travel.

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An Example

• A car with a maximum acceleration of 5.0 (km/h)/s
  is traveling at 5.0 m/s. How long will it take
  for the car to reach the speed limit of 25.0 m/s?

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Example Answer

• 14.4 sec

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Interpreting Motion Graphs

• Lets take an assessment test
• Functional Understanding of Equations

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Why The Emphasis On Interpreting Graphical
Information?

• A great deal of information can be conveyed in a
  compact form using a graph. However, if you do
  not know how to interpret the information
  displayed in the graph, that information is not
  useful to you
• In a given day, you probably encounter many
  graphs. Developing the skills necessary to
  interpret graphical information will be useful in
  your everyday life as well as for this course.

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Graph Of Position vs. Time

• Suppose we were to plot the position of an object
  as a function of time. If we are interested in
  knowing about the motion of the object between
  some initial time, ti, and some later time, tf,
  we might indicate these two points on the graph
  as P1 (when the object is at xi) and P2 (when
  the object is at xf) respectively. If we then
  join the points P1 and P2 by a straight line

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Graph Of Position vs. Time
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Graph Of Position vs. Time

• We would find that the slope of the line joining
  the two points on the position vs. time graph
  gives the average velocity of the object between
  the two instants in time.

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Graph Of Position vs. Time

• In many circumstances, however, we are interested
  in how fast an object is moving at a particular
  instant in time (as opposed to how fast it
  averaged over an interval of time). How do you
  get the velocity of an object at an instant in
  time? The process is explained in the next
  slide.

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Graph Of Position vs. Time

• This limiting process that gives the
  instantaneous velocity should be familiar to you,
  as you should have encountered it in your first
  calculus course. The limiting process yields a
  derivative

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Graph Of Velocity vs. Time

• We could instead plot the velocity of an object
  as a function of time and then indicate the
  points when the object has a velocity vi (at a
  time, ti) and a velocity vf (at a time, tf) by P1
  and P2 respectively. Then, joining P1 and P2 by
  a straight line

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Graph Of Velocity vs. Time
P1
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Graph Of Velocity vs. Time
We would find that the slope of the line joining
the two points on the velocity vs. time graph
gives the average acceleration of the object
between the two instants in time.
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Graph Of Velocity vs. Time
We can extend the limiting process we discussed
when dealing with average velocity to our
discussion regarding average acceleration. The
limiting process of calculating average
accelerations over smaller and smaller intervals
of time would yield a value for the instantaneous
acceleration of the object
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Comment About Rates Of Change

• The phrase rate of change should become
  synonymous in your vocabulary with the
  mathematical operation of taking a derivative
  (finding the slope). For example, we defined the
  velocity of an object as the time rate of change
  of its position. The mathematical definition of
  velocity we then came up with was

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Comment About Rates Of Change

• Graphically, a rate of change is found by
  calculating a slope.
• An average value is defined over a finite
  interval of time. To determine an average value
  graphically, calculate the slope of the straight
  line connecting the points corresponding to the
  endpoints of the time interval.

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Comment About Rates Of Change

• An instantaneous value is defined at a particular
  instant in time. To determine an instantaneous
  value graphically, draw a straight line that
  touches the curve only at one point (this point
  corresponds to the instant in time you want to
  know the value). Then, calculate the slope of
  this tangent line to the curve. This value is
  sometimes referred to as the slope of the curve
  even though it is actually the slope of the line
  tangent to the curve.

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Relationship Between Position and Acceleration

• If the acceleration of an object is the time rate
  of change of its velocity, and the velocity of
  the object is the time rate of change of its
  position, then the acceleration of an object is
  the time rate of change of the time rate of
  change of its position. A single rate of change
  corresponds to a single derivative, or slope.
  So, what does a rate of change of a rate of
  change correspond to?

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Relationship Between Position and Acceleration

• A rate of change of a rate of change corresponds
  to a derivative of a derivative (or more simply,
  a second derivative). If the slope of the slope
  is changing, you get a curve that bends upward or
  downward depending on the direction of the
  changes. Graphically, a rate of change of a rate
  of change corresponds to the concavity of the
  curve. A curve that has a positive concavity
  bends concave up one with a negative concavity
  bends concave down.

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Concavity

• How can you remember that concave up is a
  positive concavity and concave down is a negative
  concavity? I use pictures to help me remember.
  Put eyes over the curve

Smiley face positive person
Frowny face negative person
Positive concavity
Negative concavity
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Position vs Time Graphs Revisited

• You can determine the direction of the
  acceleration of an object at an instant in time
  by looking at the concavity of the position vs.
  time graph of that objects motion.
• Analytically, the relationship between
  acceleration and position is

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Graphical Interpretation -- Summary
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Assignment

• Begin Physlet Ch 1-2.http//www.highpoint.edu/at
  itus/physlet workbook/
• More Handouts on Graphing and its analysis
• Read Chapter 3