Superlesson
Project 4-3
Advanced Algebra
Chapter 4, Patterns and Structure in Algebra
Square roots of positive numbers have wide applications in science. Square roots of negative numbers lead to the new and exciting field of fractal geometry. You'll explore both types of square roots below.
Part B, Rational and Irrational Numbers
1. Refer to the picture below. An object propelled upward at low initial
velocity from a point A on a planet's surface will fall back to the surface,
point B. If the initial velocity is increased sufficiently, the object will
still fall, point C, but its descent path will exactly match the curvature
of the planet. Such an object will go into orbit around the planet. The
initial velocity required to achieve orbit is called orbital velocity.
If the initial velocity is increased even more, the object will escape from the planet's gravitional field altogether, point D. The required initial velocity is called escape velocity. Escape velocity v (km/sec) for a planet of mass m (kg) and radius r (km) is given by
v = 0.000 000 000 37
a. Find the mass and radius of the earth and the moon. (Masses are written for easy input into a scientific calculator. To input 4.6e8, press
. Instead of
, your calculator may have
.)
b. Find the velocity that was necessary for NASA Apollo missions to escape the earth's gravity in order to travel to the moon.
c. What escape velocity did astronauts on the moon have to achieve in order to return to earth?
d. Jupiter's mass is about 25,000 times the mass of the moon. Jupiter's radius is about 40 times the radius of the moon. Show how you can estimate Jupiter's escape velocity in comparison with that of the moon, without actually calculating Jupiter's escape velocity.
Part D, Graphing Complex Numbers
2a. Let f (z) = z2 0.4 + 0.2i. Find f (z) for z = 1 + 1.6i.
b. Find f (z) for z equal to the value of f (z) that you found above.
c. In 2a and 2b, you found two iterates of a complex function. If you were to graph many such iterates, you would create a design of gradually increasing intricacy. Computers can graph huge numbers of iterates almost instantly. The results are the beautiful and complex designs we know as fractal images. See "Mandelbrot Explorer," an interactive fractal image that you can explore in depth. Notice that the size of the complex plane shown is indicated below the image: the real axis x extends from 1 to +0.5, the imaginary axis y from 1.25 to +1.25.
d. Choose a zoom factor of "ZoomIn X 4." Click on the fractal image at a point you would like to explore in more depth. In a few seconds the computer will show you the portion of the image you clicked on, now at 4 times magnification. Notice that the new endpoints of the real and imaginary axes are given below the image.
e. Choose a zoom-in factor and click on a new point. Continue in this manner, displaying a portion of the image at increasing levels of magnification. In general terms, describe what you see.
f. Click on Mandelbrot Explorer Gallery to view other fractal images that have been created using the "Mandelbrot Explorer."