,:Ir HOW FAST CAN YOU WATCH?
by Joe Dan Austin in the Mathematics Teacher
Have you ever been riding in a car and noticed that as you pass
an object it is often not possible to follow it continuously with
your eyes? Of course, it is easy to follow far away objects as they
seem to move past slowly. However, when an object is too close to
the car, it seems to go by too rapidly for the eyes to follow.
Consider the mathematics of this phenomenon (this is a RELATED RATE
PROBLEM).
In the figure below, a car is shown on a straight road. Assume
the car travels at a constant speed r. The object being watched, a
tree, is a distance s from the road. The distance a car must travel
in order to pass by or be opposite the tree is x. A right triangle
is drawn. Using the Pythagorean Theorem, find the length of the
hypotenuse. Let 9 be the angle shown. As the eye follows the tree, the rate
that the eye moves is d9/dt, the change in 9 with respect to time t.
As the car passes the tree, the speed the eye must move in order to
follow the tree is d9/dt, for x=O.
To explain why it is not always possible to follow an object
with your eyes, we first examine d9/dt, the rate your eyes would
have to move. From the right triangle,
(I) cos 9 = Differentiating implicitly with respect to t, we find
(2)
From the triangle, we know that
(3) sin 9 =We also know that
(4) r = dt