Falling Balls. Names: Date: About this Laboratory

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1 Falling Balls Names: Date: About this Laboratory In this laboratory,1 we will explore quadratic functions and how they relate to the motion of an object that is dropped from a specified height above ground level. If air resistence and a few other factors are ignored, the height of an object may be calculated by knowing its initial distance above the ground, s 0, its initial velocity, v 0, and the time in seconds, t, after being dropped or thrown. All answers in decimal form should be to the nearest.1. ****************************************************************************** The height in feet of an object t seconds after being dropped or thrown into the air with an initial velocity of v 0 feet per second from an initial height of s 0 feet above ground level is given by the equation: s ( t ) = -16 t 2 + v 0 t + s 0. For this equation describe the following: Be as specific as possible. (You can answer the following by carefully reading this paragraph). s(t) is: The height of the object t seconds after being dropped or thrown. t is: s 0 is: v 0 is: If s( t ) = -16 t 2 + v 0 t + s 0 is graphed, which symbol is associated with the x-axis? (In other words, what is the independent variable?) Which symbol is associated with the y-axis? ****************************************************************************** Enter 41 into the white rectangle in the applet next to the word Ht., and enter 5 into the white rectangle in the applet next to the t.; then choose Set. Make sure that the Metric checkbox is not selected. Select Record. What is the equation you will use in this first example? (You should only use operations with numbers and the variable t. s( t ) = WV Transitional Math: Falling Balls- Pyzdrowski 10/2/2009-1

2 Examine your equation and then circle the type of graph that represents the equation. Line Parabola Cubic Circle Why did you choose your selection of graph type? Select the horizontal slide bar or the left and right arrow keys to move the ball. Release the button and choose Record. Repeat this process until you have at least 10 entries in the applet table. Try to find the minimum time that it takes for the ball to reach the ground as you record values. Once you have at least 10 entries, copy your values into the given table. Notice that one entry should have zero as the Time. Only record in the table for Height of zero the time that it took for the ball to reach the ground. Time Height 0 0 How can you use the table to estimate the intercepts? WV Transitional Math: Falling Balls- Pyzdrowski 10/2/2009-2

3 In the applet, select Graph and then Equat. In the rectangular box next to Y1, enter the right hand side of your equation (in terms of t). HINT: You MUST use the * symbol for all multiplication. For instance, 2(3t +5) must be entered as 2 * (3 * t + 5). Graph Once your equation is complete press the ENTER key and then select OK. You may need to Select the Out button a few times to get a better view. Sketch your graph in the space provided. Label the axes with the appropriate terms. (Use words such as length, time, height, volume,... ) What is the domain for the equation of the function? What is the range for the equation of the function? (Zoom in graphically to approximate an answer and then come back later to verify your results.) The following zoom in technique is similar to making a zoom box using a graphics calculator. 1. Move the arrow inside the graph window. 2. Position the arrow to an area slightly above and to the left of the point. 3. Press and hold the left mouse button and drag the zoom frame to an area slightly below and to the right of the point. Release the mouse button. 4. The graph window should now display the selected region. 5. Continue to zoom in until an acceptable view is obtained. To Read the Coordinates of the Point: 1. Point to and select the point of intersection. 2. Record the displayed coordinates which appear at the bottom left of the Grapher. 3. Zoom back out. Zooming Back Out: 1. Select the Back button to retrace your steps one at a time. 2. You may also select the Home button on the Grapher to return to the computer s default initial setting. What t values make sense for the problem? (This is the restricted domain. You might wish to zoom in graphically to approximate an answer and then come back later to verify your results.) WV Transitional Math: Falling Balls- Pyzdrowski 10/2/2009-3

4 What y values make sense for the problem? (This is the restricted range. You might wish to zoom in graphically to approximate an answer and then come back later to verify your results.) Zoom in graphically to approximate each t-intercept and then come back to confirm your results. x-intercept: (, 0) x-intercept: (, 0) Find the exact value of the x-intercepts algebraically by using the equation. Do not give your solution as a decimal. What do the values of the coordinates of the x-intercepts mean in this real life problem? (To answer the question, THINK: What is so important about these coordinates? What do they mean in terms of your units?) Now move the cursor and select the vertex of the parabola on your screen. Zoom in graphically to approximate the coordinates of the vertex to within.01 and then come back to confirm your results. (, ) Confirm the coordinates of the vertex algebraically by using the equation. WV Transitional Math: Falling Balls- Pyzdrowski 10/2/2009-4

5 Describe the real life meaning of the vertex of the parabola. (THINK: What do the coordinates mean in terms of your units? Where is the vertex in relationship to the other points on the graph?) Challenge: If a target were placed so that it was five feet off the ground, use algebra to determine how long it would take for the ball to hit the target? (Assume that your aim is excellent!) Hint: You can use the applet to make a good estimate, but give a solution using a fraction in its simplest form. WV Transitional Math: Falling Balls- Pyzdrowski 10/2/2009-5