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Chasing a Dodge Neon

A Dodge Neon speeds through a red light traveling at a constant speed of 75 kilometers per hour. A cop who just happened to be stopped at the light immediately begins to chase it. The police car has an acceleration of 3.2 meters per second2.

Question Group #1
Directions and/or Common Information: 

Your first task is to make a graph of the displacement versus time for both vehicles. From this graph you will be able find the time required for the police car to catch up to the Neon. Follow the 4 steps listed below to set up your graph. You will not need a graphing calculator until Step 4.

  1. Write the equations for the displacement of both vehicles:


    • The Neon is traveling at a constant velocity. Its equation will be s = vt
    • The Police, on the other hand, are experiencing uniformly accelerated motion. Their equation will be s = v0t + ½at2


  2. Substitute out the variables s and t with the graphing variables x and y:


    • Neon: y = vx
    • Police: y = v0x + 0.5ax2


  3. Now you must replace the remaining variables with their proper values. There is a slight problem with this because the speed of the car was measured in kilometers per hour, whereas the acceleration of the police car was given in meters per second2. To attain consistent units you must either convert 75 kilometers per hour into meters per second, or convert 3.2 meters per second2 into kilometers per hour2. For this exercise, the first conversion will be easier and more practical.
    First, divide 75 kilometers per hour by 60 minutes. This gives 1.25 kilometers per minute. Divide by 60 a second time. This gives 0.02083 kilometers per second. Finish by multiplying that value by 1000. The end result should be 20.83 meters per second.
    Now you may substitute into the formula:


    • Neon: Y = 20.83x
    • Police: Y = (0)x + 0.5(3.2)x2


  4. You should now enter these equations into the calculator. Do this by pressing , and then type the equations in the Y1 and Y2 fields. (Note: Y= is already entered, and cannot be changed). To enter an x, you can either press then , or press . After you enter the first equation, either press down or enter to go on to the next field. To square the third x, press immediately after entering the x (Fig. 4.1).


    Fig. 4.1

  5. Press . As you can see (Fig. 5.1), the data does not fit properly on the screen.


    Fig. 5.1

    To correct this, press (Fig. 5.2).


    Fig. 5.2

    The values entered here define the region displayed on the graph. A good window is usually found by estimating the maximum values likely to be useful, aided by a little playing around. Enter 0 for Xmin, 20 for Xmax, 0 for Ymin, and 400 for Ymax (Fig. 5.3).


    Fig. 5.3

    Press again. Now you can see the graph of displacement versus time for the police car and the Neon, and the two locations that they overlap (Fig. 5.4).


    Fig. 5.4



What is the graph telling you when you see that the two curves intersect at two positions?
1. 




Question Group #2
Directions and/or Common Information: 

Now that you have your graph, your second task is to find the time required for the police car to catch the Neon. To do this, press enter then . This will take you to the “Calculate” menu. Select item 5 and press Enter (Fig. II.1), or just press 5.


Fig. II.1

It will take you to the graph, with a crosshair on the line (Fig. II.2).


Fig. II.2

Press enter to confirm the line as the first “curve”. The crosshair will now be on the parabola. Press enter again to confirm the parabola as the second “curve”. The calculator will now ask if you would like to guess. Press enter without typing anything to bypass this. The crosshair will now center on the intersection of the line and the parabola at (13.01875, 271.18056) (Fig. II.3).


Fig. II.3



What does the x value of the intersection found in Fig. II.3 represent?
1. 




Question Group #3
Directions and/or Common Information: 

Your next task is to find the speed that the police car had to attain to catch the Neon. Again, press . This time, select item 6 (Fig. III.1).


Fig. III.1

You will be taken back to the graph, with the crosshair over the intersection again. Notice that in the top left-hand corner of the screen, the equation of the line is displayed. This area of the graph always displays the equation of the item currently selected. Press either up or down to select the parabola (Fig. III.2).


Fig. III.2

Now press enter to confirm the parabola as your selection. The calculator will now display the speed attained by the police car, 41.531915 meters per second (Fig. III.3).


Fig. III.3



How fast was the Neon traveling when the Police car caught it?
1. 




Question Group #4
Directions and/or Common Information: 

After the court hearing, the driver of the Neon still does not change his driving habits; instead, they get worse. One month later, he speeds through the same intersection - this time going 120 kilometers per hour. Surely, no police car would be able to catch up with him from a dead stop before he could get out of sight. Unfortunately for him, there were three police motorcycles waiting there, having just gotten off their coffee/doughnut break.


If each these motorcycles has an acceleration of 5 meters per second2, how much the time would be required for the police to catch the car? How fast would the motorcycles be traveling?
1. 








J Burch

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