The National Council of Teachers of Mathematics has placed a lot of emphasis on using multiple representations when teaching mathematics. By that they mean students should be able to deal with mathematics by writing about it, talking about it, viewing it visually such as when graphing or as in geometric drawings, and by viewing it algebraically as in equations and functions.
One of the classic problems in first year physics is to calculate the amount of bank needed to prevent cars from sliding off the highway when they round a curve at a given speed. This problem can benefit from multiple representations.
This problem comes up in my AP Physics C course from Monterey Institute, it is problem 2 part b) on the end of Chapter 5 FRQ. Here is how it is stated there:
In order to drive a car around a curve, there must be a frictional force between the tires and the road, or the road must be banked. Consider a 1250 kg car traveling at a speed of 25.0 m/s around a curve with a radius of 175 m.
b.) If the curve is banked and the road surface is frictionless, what must be the angle (with respect to the horizontal) of the road surface?
If you search the Internet, you will also find a good drawing and a good explanation of the solution to this problem at the Batesville.k12.in.us web site , explaining why the relationship between the velocity, the radius of the curve, the acceleration of gravity, and the need angle will be given by:
And therefore the needed angle will be given by:
In addition to looking at the problem in terms of equation, I also want to look at this problem graphically in a free body diagram, and I want to suggest how we might relate these either by asking the students to write about their solutions, or to discuss their solutions in class. When looking at the free body diagram, there is now software out there that will allow you make a dynamic free body diagram that can be manipulated to see how things change with the amount of bank in the road.
The equations themselves, and where they come from, are nicely covered at the above Batesville.k12 site. By looking at this graphically we can go farther, getting an even deeper understanding of the physics that is happening in the problem, getting a feel for the underlying mechanics.
I have added a "dynamic sketch" of the situation at http://mtl.math.uiuc.edu/~t-anders/blog/road_slope_problem.htm . (This diagram relies on your browser being able to handle Java, most newer browsers can.)
Line segment BO represents the bank on the road. If you drag point B you can change the angle of the bank on the road (angle AOB). The car is located at point G, and vector GD represents the normal force exerted on the car by the road. As you move point B, try and relate what you are looking at in the free body diagram, to what you are seeing in the Batesville equations, and to the description of the original problem. See if what you are looking at visually makes sense. Think about questions you might ask your students as they look at the multiple representations.
In the next blog, I will talk about the underlying physics, and suggest some questions you might ask your students, either in class, or in a written exercise, to cause them to think about whether a particular solution really makes sense.Content related to this blog posting can be found at HippoCampus under Uniform Circular Motion, Centripetal Acceleration, and Racetrack - Simulation.