Chapter 13. Acceleration
Volume 3, Why Slopes and More Math.
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.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
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Road
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See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
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by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
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of improper or false imprisonment, or for execution. Site chapters on Logic
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Based Reason may slowly lead to greater precision in reading, applying and
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
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McCainian: drill, drill, drill then Toronto
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Fractions for
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Mathematics
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Skills with take
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Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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