Laboratory One - Distance & Time 1 Background When an object is propelled upwards, its distance above the ground as a function of time is described by a quadratic function f(x) = ax2 + bx + c. In this lab you will roll a ball up a ramp. You will make use of the CBL and a motion detector to measure the distance along the ramp that the ball is above or below a fixed reference point. The CBL will take distance measurements at equal intervals of 1 100 of total time you enter. For example if you enter 5 seconds for the time, it will take 100 measurements. One every 0.05 seconds. It will send the measurements to the calculator which will convert it to data, which you will see in graph form. The pull of gravity and friction will slow the ball as it rolls up the ramp until it comes to a stop and then rolls back down, speeding up on its way. In this lab you will use the fact that this distance, as a function of time, will be a quadratic function. You will attempt to find the exact formula for the function which models the graphically represented data collected by the CBL and displayed on the calculator when you roll the ball up the ramp. 2 Lab The table will be set up as a ramp with the motion detector and CBL unit ready to go. One of our calculators will be connected to the CBL unit. It will have the program RAMP already installed. You will use this calculator to collect the data and then transfer the data to your calculator. Step 1: Practice rolling the ball slowly up the ramp so that it passes the second line (marked Zero Here), but not the last line. Have one student ensure that the ball does not hit the motion detector. Estimate the number of seconds it takes for the ball to roll up the ramp and back down to you. You will enter this estimate into the calculator later. Step 2: If necessary, turn the calculator on, and then press the PRGM button. Use the arrow keys to scroll down to the program called RAMP then press the ENTER button twice. You will see the introductory screen. Press ENTER then press ENTER again and follow directions on your calculator. Step 3: Now, following the directions on your calculator, press ENTER . Enter your estimate of the number of seconds from your practices in Step 1 and press ENTER . Place the ball next to line marked Zero Here and press ENTER while holding the ball still. This establishes the fixed reference point. Return the ball to the bottom of the ramp. Press ENTER . Have one student count down and press ENTER , starting the program just slightly before another student starts the ball rolling up the ramp for its return under gravity. 1 Step 4: You should now have a plot (graph) of data on your calculator. Does it look like a parabola? If not, run the program again beginning with Step 2. If part of the parabola is cut off or if there is a flat line at the right side of the parabola, adjust the time. If there is a flat top on the curve, the ball got too close to the motion detector. Also, there must be two x-intercepts. Repeat steps 2 through 4 until you get a nice parabola. Step 5: The lab assistants will help you transfer the data to your calculator in order for you to analyze the data and complete the lab report. 3 Analysis of Data Now you are ready to find the function that described the position of the ball from the first marker as a function of time. In this case, x will represent the time; f(x) or y will represent the distance from the first tape. Remember, we are saying that the distance will be described by a quadratic equation (i.e. f(x) = ax2 + bx + c) for some constants a, b, and c. Your job is to determine a, b, and c. Part I: Trial and Error Intercept Approach The factored form for a quadratic function is f(x) = a(x - x1)(x - x2), where x1 and x2 are the x-intercepts. Trace your data plot to estimate the x-intercepts. x1 = x2 = . Now you are going to find a parabola that has these x-intercepts, namely f(x) = (x - x1)(x - x2). f(x) = Enter this function under Y1 in factored form in your calculator and graph it. How does your graph compare to your data plot? Does it have the same x-intercepts? You will need to multiply the entire expression by a constant (a) until the graph is a very close fit to your data graph (i.e. f(x) = a(x - x1)(x - x2)). Will your value for a need to be positive or negative? Does the graph from your equation need to be flatter or steeper? Will your constant need to have absolute value smaller or larger than 1? 2 By trial and error, find the value of a that provides the best fit. What value of a provides the best fit? What is your equation in the form f(x) = a(x - x1)(x - x2) found by trial and error? f(x) = Vertex Approach The vertex form (or standard form) for a quadratic function is f(x) = a(x - h) 2 + k, where (h, k) is the vertex. Trace your data plot to estimate the vertex as closely as possible. The vertex is . Enter your function in Y2 as (x - h) 2 + k. Turn Y1 off in your calculator so that it doesnt graph. How does your graph compare to your data plot? Does it have the same vertex? Does it have the same x-intercepts? You will need to multiply the entire expression by a constant until the graph is a very close fit to your data graph (i.e. f(x) = a(x - h) 2 + k). Will your value for a need to be positive or negative? Does the graph from your equation need to be flatter or steeper? Will your constant need to have absolute value smaller or larger than 1? By trial and error, find the value of a that provides the best fit. What value of a provides the best fit? How close does the graph of the equation you found using the vertex match the graph of the equation you found using the x-intercepts? 3 Part II: Algebraic Exploration Intercept Approach Using the same values for x1 and x2, assume f(x) = A(x - x1)(x - x2) for some constant A. Trace your data graph to find one additional data point (x3, f(x3)). (x3, f(x3)) = Using the values for (x3, f(x3)) in the equation, f(x3) = A(x3 - x1)(x3 - x2), solve for A. Show your work. A = . Using this value for A and your x-intercepts, what is your equation in the form f(x) = A(x - x1)(x - x2) found algebraically? f(x) = How closely does this value for A agree with the one you determined by trial and error in Part I using the intercept approach? Enter this function under Y3 and graph it to determine if the function closely models the data points. Which value of A (the one determined by trial and error or the one determined algebraically) produces the graph which most closely models your data point graph? Using your best value of A, multiply your equation through and put it in the general form f(x) = ax2 + bx + c. f(x) = Vertex Approach Using the same values for h and k, assume f(x) = A(x - h) 2 + k for some constant A. Trace your data graph to find one additional data point (x3, f(x3)). (x3, f(x3)) = 4 Using the values for (x3, f(x3)) in the equation, f(x3) = A(x3-h) 2+k, solve for A. Show your work. A = . How closely does this value for A agree with the one you determined by trial and error in Part I using the vertex approach? Enter this function under Y4 and graph it to determine if the function closely models the data points. Which value of A (the one determined by trial and error or the one determined algebraically) produces the graph which most closely models your data point graph? Using your best value of A, multiply your equation through and put it in the general form f(x) = ax2 + bx + c. Compare the two equations in general form that you found algebraically. How close are these two equations? Lastly, plot at least 5 points of the original data curve. Be sure to label the axes, x-intercepts and vertex. ? ? ? ? 5