Estimating the Instantaneous Speed of a Moving Object
A ball is rolling down an inclined plane. The distance it rolls is
given by the formula
where s represents the distance it has rolled t seconds after it
starts rolling. Your task is to estimate
how fast it is moving
2 seconds after it starts rolling.
You can use the given formula to find the average
speed over a time interval. For example, you
can find that s=5*23
=40 when t=2, so it rolls 40 feet in
the first 2 seconds and thus has moved at an average speed of
40/2=20 feet per second during that time interval.
Similarly, since s=5(33)=135 when t=3, the ball rolls a distance of
135-20=115 feet over the 1 second time interval 2 ≤ t ≤ 3 and
thus rolls at an average speed of 115 feet per second over that time
Your problem, however, is not to calculate an average speed over
an interval of time, but
to estimate the instantaneous speed at an instant in time. That
will involve analyzing what is meant by instantaneous speed.
The class will break up into two groups of three or four students. The
groups will have thirty (30) minutes during class to analyze the
question. Each group will then briefly share its analysis with the
rest of the class. Comment from the rest of the class is appropriate
during these presentations.