# Lesson Plan

 The Problem: The driver of a car travelling 60 mi/hr (miles per hour) crests a hill and suddenly sees that traffic is stopped approximately 88 feet further down the road. So, the driver slams on the brakes causing the car to slow down at a rate of 49090.9 mi/hr2. The driver deperately needs to know the answer to three questions: How long will it take this car to stop? How far will the car go before it stops? Will the car stop before it crashes into the cars ahead? The Way to the Answer: The following steps will lead you to an answer and teach you the graphical relationship between integration and differentiation: Convert the given quatities to ft/sec (feet/second) and ft/sec2. Why?    You'll probably want your answer to be in feet (as opposed to some small fraction of a mile) and seconds (instead of a little fraction of an hour). Plot points reflecting the area under the deceleration graph for various times.    The value you use for deceleration will be the value you found in step 1 which is in units of ft/sec2. Estimate a smooth curve passing through the points obtained by step 2.    This curve's graph will estimate the car's velocity versus time graph, and it's equation estimates it's velocity as a function of time. Read off the graph obtained in step 3 the time at which the car's velocity is 0.    This is the time it takes the car to stop. Plot points reflecting the area under the graph of step 3 for various times. Estimate a smooth curve passing through the points obtained by step 5.    This curve will estimate the car's distance versus time graph, and it's equation estimates it's distance as a function of time. At the time that the car's velocity is 0 (which was determined in step 4), calculate the distance with the distance equation you determined in step 6.    This is the distance it takes the car to stop. Do you think your answer is correct? This step is the all important check step.    Upon twice differentiating the equation of the car's distance as a function of time, you should approximately get the car's given deceleration. It will only be an approximate since you only approximated curves, and areas under curves, when finding this equation. If your value is not close enough to the given value, then go back to step 2, take better extimates of areas and try to fit your curves better. Finally, the answer to this problem will be determined analytically. Next: Follow the "Convert Quantities" link to the left.