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  13 Acceleration  Back ] Home ] Next ]    

Chapter 13
Acceleration

Acceleration As a Slope of A Slope

 

A graph d = f(t) of distance versus time is shown in the first graph. The second graph shows the graph of the velocity v = f¢(t), that is, the slope of the first (distance) graph, versus time. The third graph maps the slope a = f¢¢(t) of the second (velocity) graph versus time. This second slope is called the acceleration.


FOOTNOTE: The word derivative could and perhaps should be used here in place of the word slope.

At time t1, the acceleration is greatest. The rate of change of velocity before and after is less. At time t2, the velocity changes sign and the direction of travel changes as well from negative to positive. The distance to the origin is a minimum at t2.

Another Varying Velocity Example


The direction of travel changes and the sign of the slope, here velocity, changes at times t = t2 and t = t4. The slope or velocity is greatest at the times t = t1 and least at the t = t3. These two times yield inflection points on the distance versus time graph. These points identify the times when the speed (= the magnitude of the slope on the distance versus time graph) is greatest and least.

Our physical sense of speed and velocity corresponds to the measurement of slope and the behavior of this slope on a distance versus time graph. Thus driving or riding on a bicycle or in a car, our physical senses indicate when the speed is changing in magnitude or direction. In a car or on a bicycle, a driver or rider can sense where the speed or velocity

  • is positive for forward motion;
  • is negative for backward motion;
  • zero for no motion or not moving;
  • increasing, that is, accelerating for forward motion and decelerating for backward motion; and
  • decreasing, that is, decelerating or braking for forward motion and accelerating for backward motion.

As slope or velocity increases, the forward motion accelerates. When velocity decreases the forward motion decelerates. In car or bicycle motion, acceleration comes from applying enough energy to overcome rolling resistance while deceleration comes from a lack of such energy or from deliberate braking.

Varying UniDirectional Velocity


The first diagram above shows the graph of distance versus time. The slope of this first graph versus time is plotted in the second diagram. The slope with its units of distance over time has the previously mentioned interpretation of speed or velocity. Here the speed (slope) in this graph remains positive, but it increases and decreases. See the second graph. Physically, this corresponds to a unidirectional, forward motion which is subject to acceleration and deceleration, that is speed increases and decreases.

Why Slopes
and
More Math

understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7

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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc


If  you like Volume 3  you may also like  Three Skills for Algebra , Exponents & Radicals Exactly,  complex numbers, Euclidean Geometry , More Calculus and  Pattern Based Reason  as well. 

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity


These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem
is a postscript,
not in printed version.


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