Microworld: Calculus
in Action: Chapter 1, Section 1: Ballistics
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Author:James
E. White
Calculus
is a habit of thought. At the heart of that habit is the notion of approximation.
The derivative is essentially a means of approximation, and by means of it,
you will learn when and how approximations may be useful, and may even lead
(somewhat paradoxically) to exact results! But another and older method of
approximation was available long before the Calculus was invented. And Galileo
used this method, which was due to Archimedes, to calculate the exact distance
fallen by a uniformly accelerated object. This method will be called integration
in the Calculus.
Ballistic
trajectories according to Galileo
Recall
from Natural Motion and Uniform Acceleration, the introductory microworld
in these lectures, the picture of motion of a freely falling object that we
shall call the velocity portrait. You see that picture at the top of
this page.
In
this picture, the successive yellow rectangles have the same width (a fixed
duration of time) and their heights represent the speed of the ball
at the end of the interval. Since this speed changes continually, the area
of each rectangle is only an approximation to the actual distance fallen
during the interval. Also as you can see, the speed changes in a uniform
way. We might say that there is a positive constant G such that at
the end of the ith interval, the speed is G*i . This is Galileo's velocity
portrait of a uniformly accelerated motion.
As
you read in the Introduction, Galileo believed that if O is any freely falling
object and height(t) measures its height in meters at time
t, measured in seconds, then its vertical velocity
v(t), is
what is directly affected by gravity.
In
particular, Galileo asserted that the effect of gravity on such an object
(whatever its weight) was to cause equal increments in downward velocity
over equal intervals of time. We will in all of these exercises,
ignore the effect of air resistance as Galileo did. Thus, for a projectile,
v(t) is a linear function of time.
Now
Galileo argued that if the true velocity portrait is a triangle rather than
a staircase, then the true distance fallen over the interval of time must
be the area of that triangle. To calculate the area of the triangle, we have
to know what v(t) is as a linear function of time. Galileo
argued that (given choices for units in time and distance) there is a constant
g such that in fact v(t) = g*t.
In
this microworld, we will experiment with this assertion while we attempt to
answer several questions about projectiles.
Question 1: Assume that a projectile is launched vertically at an initial height H with an initial vertical velocity V. Write a formula that expresses the height: height(t) as a function of time, t.
Question 2: Suppose the initial speed V was given as constant, and the initial height is 10 meters. And suppose you could "adjust" the angle A.
- Write a formula that expresses as a function of A the greatest height, attained.
- Write a formula that expresses as a function of A the amount of time, the projectile is in the air.
- Write a formula that expresses as a function of A the horizontal distance traveled before the projectile returns to the ground.
Question 3: Next, we would like to determine which angle will cause our projectile to travel the greatest horizontal distance. As a preliminary, we will solve this problem when the initial height is 0. When the height is 10 meters, the problem is much more difficult. We will develop an approximate method to solve this problem in Section 3 of this Chapter, after we discuss Newton's approximation Method.
The
following is the Table of Contents for this 4 page Microworld
- Section 1: Finding the Maximum Height
- Section 1: Finding the Maximum Range
- Symbolic Calculator
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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