Laboratory One Distance and Time

Transcription

1 Laboratory One Distance and Time Student Laboratory Description Distance and Time I. Background When an object is propelled upwards, its distance above the ground as a function of time is described by a quadratic function d f() t at bt 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 of total time you enter. For example if you enter 5 seconds for the time, it will take measurements every 0.05 seconds. It will send the measurements to the calculator which will convert it to data 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 II. 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. Practice rolling the ball slowly up the ramp so that it passes the first blue tape but not the second, white (masking) tape. 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. 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 and press ENTER. Place the ball next to the first (blue) tape 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. 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. 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 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 Student Laboratory Worksheet---Distance and Time Now you are ready to find the function that described the position of the ball from the first tape as a function of time. Now, 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., constants a, b, and c. Now the job is to determine a, b, and c. ( ) = + + for some f x ax bx c Part I: Trial and Error Intercept Approach Trace your data plot to find the x-intercepts as close as possible. Denote the x-intercepts as x and x. x = x = Now you are going to find a parabola that has those same x-intercepts, namely f ( x) = ( x x)( x x). f(x) =. Enter your function under Y in factored form in your calculator and graph it. i.e. Y = ( x x)( x x) 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 constants until the graph is a very close fit to your data graph i.e. f ( x) = k( x x)( x x). Will your value for k 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?. By trial and error, find the value of k that provides the best fit. What value of k provides the best fit?. What is your equation in the form f ( x) = k( x x)( x x) found by trial and error?.

4 Vertex Approach The vertex form for a quadratic function is f ( x) = a( x h) + k, with vertex (, ) h k. Trace your data plot to find the vertex as close as possible. The vertex is. Enter your function in Y as ( x h) + k. Turn Y off in your calculator so that it doesn t 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 + 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?. 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 h + k found by trial and error? 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?

5 Part II: Algebraic Discovery Intercept Approach Now, try to find your equation for f(x) algebraically as follows: Using the same values for x and x again assume f ( x) = K( x x)( x x) for some constant K. Trace your data point graph to find one additional point ( x3, f( x 3)). It is. Using the values for x 3 and f ( x 3) in the equation, f ( x3) = K( x3 x)( x3 x), solve for K. Show your work. K =. The equation is f(x) =. How closely does this value for K agree with the one you determined by trial and error?. Try this value of K in your function Y = K( x x)( x x) and graph it to determine if the function closely models the data points. Which value of K (the one determined graphically by trial and error or the one determined algebraically) produces the graph which most closely models your data point graph?. Use your best value of K. Multiply through, and you will have your function in the standard form f ( x) ax bx c = + +. It is. Vertex Approach Now, try to find your equation for the vertex form of f(x) algebraically as follows: Using the same values for h and k again assume ( ) ( ) f x = A x h + k for some constant A. Trace your data point graph to find one additional point ( x3, f( x 3)). It is. Using the values for x 3 and f ( x 3) in the equation, f ( x3) A( x3 h) k = +, solve for A. Show your work. A =. The equation is f(x) =. How closely does this value for A agree with the one you determined by trial and error?. Try this value of A in your function ( ) f ( x) = A x h + k and graph it to determine if the function closely models the data points. Which value of A (the one determined graphically by trial and error or

6 the one determined algebraically) produces the graph which most closely models your data point graph?. Use your best value of A. Multiply through, and you will have your function in the standard form f ( x) ax bx c = + +. It is. Compare the two equations you found in standard form algebraically. How close are these two equations? Sketch below a graph of your data points with the axes, labeled and scaled. Label at least five points.