Substituting (3) and (4) in (2), and then solving for de/dt,
we get
de
(5) = dt
Since we assumed that both r and s are constant, the maximum for
de/dt occurs when x=O:
(6) So de/dt = Now we can explain why it is difficult to follow some objects as
you pass them. As the rider watches an object that is close to the
car, the value of s is small. The specific value of s can be made as
small as we wish by choosing an object sufficiently close to the
road to watch. From (6), this means that the maximum rate r/s that
the eye must turn in order to follow the object can be made larger
than any finite value. Specifically, r/s can be made larger than
the rate that the eyes can move.
Let's estimate how near an object can be to the road and still
be followed with the eyes when the car is driven at the current
speed limit of 55 miles per hour (80.67 feet per second). To
determine how fast your eyes can move and follow an object, hold
your arm straight out and watch your thumb. Move your arm and thumb
(but not your head) as fast as you can and still follow the thumb
continuously with your eyes. Move the thumb a full 180 degrees or
~ radians. Suppose you can do this in 1/2 second. The maximum
angle movement in radians for your eyes would be
(7) = radians per second.
Using (6), you can follow the object as you pass it if
(8) r/s <.
For a car traveling 55 mph, r = 80.67 feet per second. Using this
value in (8) and solving for s implies that you can follow the
object as you pass by it if the distance s of the object from the
road satisfies
r
(9) s » = feet
~
21t'
Thus, if the object is closer than feet, your eyes
can NOT continuously follow it as you pass it.
Try this experiment the next time you are riding (NOT driving).
When traveling at 55 mph, how close can an object be to the road
before you are unable to follow it with your eyes? How good is the
estimated value in (9) for you?