Estimating the Instantaneous Speed of a Moving Object
ProblemA ball is rolling down an inclined plane. The distance it rolls is given by the formula s=5t3, 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.
BackgroundYou 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 interval.
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.
Ground RulesThe 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.