Charlie gets a ticket. He is charged with having driven his Honda at a speed of 70 km/h having travelled fifty metres from a traffic light in town where only 50 km/h are allowed. The police send him a photo of a speed trap as evidence. As Charlie had to stop at a red light he believes he is able to prove that he was unable to attain the indicated speed on at this distance from the light by the help of a graph that represents the distance-time ratio under optimal acceleration conditions for his Honda. The curve in the interval from 0 to 10 seconds is approximately described by the function s(t) = 0.1 t3 + 0.75 t2 + 0.25 t Will this graph help Charlie prove his case?
Notes for teachers: The problem can be used as an introduction in derivation of a function at one point of it’s domain. The given function means the time t measured in seconds an the distance s in metres. Discussing the problem, student have to convert the speed from 70 km/h to 19,44 m/s. Then one has to find the time that belongs to the distance of 50m. Students discuss methods to find out the maximum speed at s = 50m. Often they suggest graphical methods to get the speed (i.e. slope in a point). But the parameters of the function are such that the speed at 50 m is about 20 m/s. So graphical methods dont’t allow a decision and one has to develop a method to calculate the slope of a curve in a given point by the limit of quotients. To use the given problem in a lesson, the students should know something about movement with constant velocity and they should be familiar with the concept of limits. Thanks to my colleague Helmut who helped to write in English! |