Why Slopes
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| Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master. Also includes arithmetic review problems to catch common mistakes. A fourth skill gives a unifying theme for high school maths. |
Units in Calculations:
Appendices:
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.
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.
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
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.
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.
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.