You will solve application problems involving rate and distance using a problem solving plan.
After completing this tutorial, you will be able to complete the following:
The distance traveled by an object traveling at a constant speed is given by the formula , where is the constant speed, is the travel time, and d is the distance traveled. This formula, known as the distance formula, can be derived by straightforwardly interpreting speed as a rate of change. Speed measures the rate by which distance changes with respect to time. For example, consider a car traveling at 55 miles per hour. The car moves 55 miles after one hour and an additional 55 miles for every mile thereafter.
The above table calculates distance after one, two, and three hours by adding 55 miles to the total travel distance for every additional hour. This method of determining distance is flawed for three reasons. First, it would be grossly inefficient to use this method to calculate distance for lengthy travel times; a 100-term sum is required to calculate the distance after 100 hours of travel. Second, this method provides no means for determining distance for fractional travel times, such as 1.5 hours. Third, this method provides no insight or predictive ability. These flaws compel us to seek a different method. Consider the table below:
The speed of some object relative to a given reference point is the rate by which the object's distance from the reference point changes over time. For example, the speed of a baseball thrown from a moving vehicle in the direction of travel can be measured from different reference points. If measured from the vehicle, the ball's speed is the rate at which the car moves away from the vehicle. On this account, a pitcher capable of 100 miles per hour pitches would observe that the ball moves at 100 miles per hour. Compare this viewpoint to the stationary observer who, upon measuring the speed of the ball, finds that it is traveling at 100 miles per hour plus the speed of the vehicle. So for example, if the vehicle speeds along at 100 miles per hour, the stationary observer measures the speed as 200 miles per hour.
These sorts of relative speed calculations are accounted for by Galilean relativity, the basic ideas of which should be clear to any airplane passenger who, while impatiently waiting for the food cart, somehow ignores the fact that the food cart actually barrels forward by several hundred miles per hour relative to the Earth.
Please note that to make the problems used in the Activity Object simpler to understand, it is assumed that as soon as the cars start to move, they can immediately achieve their constant rate of travel.
|Approximate Time||40 Minutes|
|Pre-requisite Concepts||Students should know how to solve one-step linear equations and two-step linear equations.|
|Type of Tutorial||Problem Solving & Reasoning|
|Key Vocabulary||constant speed, distance, distance formula|