Open-ended Questions for Automotive Airbag Example
The GE fund grant will allow integration of the existing airbag module
with interactive math and physics exercises that explore the mathematical and
physical relationships behind the rate of inflation of the airbag, as well as
the force with which the airbag and the body of the driver collide. Sample
open-ended questions the students might be asked in mathematics and physics are
(1) Curve shape analysis to determine optimum rate of gas production.
The rate at which the airbag opens is critical if it is to function properly.
The bag must be full within 100 milliseconds and must not inflate too rapidly
so as not to exert too much force on the driver's body. A graphical depiction
of the volume of gas produced as a function of time is an excellent way to have
students think about the rate of change in volume required to meet both
criteria. Students' conceptual understanding of the meaning of the plots will
be developed by having them provide a description of the appearance of the
airbag inflations from the driver's point of view in the scenarios described by
the different curves.
(2) Assuming that the airbag approximates a sphere, for each curve students
will determine the maximum velocity with which a driver would collide with
the airbag. They will need to explore the diameter of the airbag D =
D(t) and its derivative d D/ dt (t) as
functions of time in order to obtain the velocity at which the bag approaches
the driver. The volume "V" of a spherical ball is related to the
diameter by the standard formula V = ([[pi]]/ 6) D3.
Students will explore how to relate the derivatives dV/dt and
dD/dt and how to calculate one from the other. Symbolically,
this is an illustration of the chain rule and the technique of implicit
differentiation, both of which are very important topics in beginning
calculus. Furthermore, students will use their computers to explore the four
functions V, D, dV/dt, and dD/dt and
their relationships graphically and numerically by displaying their graphs,
observing how the graphs change as the parameters of the chemical reaction
change, and by computing actual values of the functions. Finally they will
relate this graphical and numerical information to the symbolic calculations.
This work is a simple but effective illustration of the basic pedagogical
principle in reformed calculus of using a concrete example to stress the
multiple ways of representing functional relationships (symbolic, graphical,
numerical, and verbal). The example emphasizes the interplay between these
different representations and how information about the function, its
derivatives and integrals is transported from one representation to another.
The use of computers in the class-laboratory/studio is absolutely essential to
achieve these pedagogical goals.
(3) Students can then examine how the airbag alters the collision process by
determining the amount the force of impact is lessened by having the driver
collide with the airbag rather than the steering wheel of the car. They can
also compute the force of impact of the exploding airbag with a stationary
driver, to determine why it is so important to avoid false triggers.
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