Module 9  Velocity  
Introduction  Lesson 1  Lesson 2  Lesson 3  Self Test  
Lesson 9.2: Instantaneous Velocity  
In Lesson 9.1 you found average velocities of a falling book over different time intervals. In this lesson you will find the instantaneous velocity of the falling book at t = 0.16. In Lesson 9.1 the average velocity of the book from t = 0 to t = 0.16 was found to be approximately 0.931 m/s and the average velocity of the book from t = 0.16 to t = 0.28 was found to be approximately 2.183 m/s. Instantaneous velocity is the velocity of an object at a specific instant, which is the number that would be read from a speedometer at that particular moment. Each of the average velocities previously found could be considered an approximation of the instantaneous velocity at t = 0.16, but neither was exact. Getting Better Approximations You can use average velocities to find a series of approximations that will lead to the instantaneous velocity of an object at a given instant by letting the elapsed time between measurements be very small. 9.2.1 Find the average velocities over shorter and shorter intervals near t = 0.16 by using the following data points. Click here for the answer.
9.2.2 Which of the three average velocities is the best approximation to the instantaneous velocity? Why? Click here for the answer. Fitting a Curve to the Data The smaller the elapsed time interval, the better the average velocity approximates the instantaneous velocity. The smallest time interval provided by the data near t = 0.16 has length 0.02. However, a regression equation that approximates the data can be used to obtain points even closer to t = 0.16 and those points can then be used to find a better approximation of the book's instantaneous velocity at t = 0.16. The shape of the data appears to be parabolic, so a quadratic equation should fit the data.
Notice the third parameter (Y_{1}) for the regression command is missing and that the calculator will not store the regression equation in the Y= editor. You will round off the coefficients of the regression equation and manually enter the rounded version in the Y= editor.
Using the Regression Equation The height of the book at t = 0.16 and t = 0.18 can now be approximated by finding Y_{1}(0.16) and Y_{1}(0.18). These values can be used to approximate the average velocity of the book between t = 0.16 and t = 0.18 by computing
The result is a good approximation of the average velocity of 1.75 m/s you found using the table data values over the same interval in Question 9.2.1. Better Approximations Points even closer to (0.16, 0.716), which represent shorter time intervals, may be found by using the quadratic function stored in Y_{1}. Find the average velocity for the interval from t = 0.16 and t = 0.17 by evaluating Instead of pressing all the keys to enter the above expression, recall and edit the last command on the Home screen.
9.2.3 Using the regression equation in Y_{1} and the method described above, find the average velocities for the smaller and smaller time intervals shown below. t = 0.16 to t = 0.161 (elapsed time = 0.001 seconds) t = 0.16 to t = 0.1601 (elapsed time = 0.0001 seconds) t = 0.16 to t = 0.16001 (elapsed time = 0.00001 seconds) Click here for the answer. Finding Instantaneous Velocity As the elapsed time approaches zero, the average velocities approach a limiting value. This limit is the instantaneous velocity and it can be found by letting h represent the elapsed time and evaluating The three average velocities found in Question 9.2.3 could also be found by letting h = 0.001, 0.0001, and 0.00001 and evaluating .
Evaluating for smaller and smaller values of h gives an approximation for , the instantaneous velocity at t = 0.16. Finding the Exact Value of the Limit In order to find the exact value of the limit, the expression must first be simplified, as shown below. Recall that Y_{1} = 4.503x^{2}  0.209x + 0.864.
The limit of the last expression as h approaches 0 is 4.503(0.32)  0.209 = 1.64996, so the instantaneous velocity at t = 0.16 is approximately 1.64996 m/s. Illustrating Instantaneous Velocity Just as the slope of the line that connects two points on the graph of height vs. time represents the average velocity between the two points, a special line is associated with the instantaneous velocity: the tangent line to the curve at the desired point. The slope of the tangent line to a curve at a given point is defined the same way as instantaneous velocity. Definition of a Tangent Line to a Curve The tangent line to the graph of f at the point (a, f(a)) is the line through (a, f(a)) with slope provided the limit exists. Drawing a Tangent Line A tangent line to a curve at a given point can be drawn with the Tangent feature, which is found in the Draw menu.
The equation at the bottom of the last screen, y = 1.64996x + 0.9792768 , is the equation of the line tangent to the curve at the point t = 0.16. Notice that the slope of this line, 1.64996, is equal to the instantaneous velocity found earlier by evaluating the limit


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