Area Under The Acceleration Graph

In this step, you will plot acceleration versus time on a graph, and then calculate the area under that graph at various times. The values of those areas will then be plotted on their own graph in the next step.

You will need to use the "Area" tool to accomplish this. Under "Tools" to the left, click on the "Area" button. A separate window, named "Estimate Area Under Curve," should pop up in which you will graph acceleration versus time and calculate the area under that curve for various times.

This tool is easy to use -- click the demo button for a quick demonstration of how to use it. If you're still having trouble understanding it, then try the "Help" button.

In the "Estimate Area Under Curve" window, plot acceleration versus time by entering -20 in the "y =" text field then pressing the "Graph Your Equation" button. On this graph, x will stand for the time variable in seconds.

Now you need to estimate the area under the graph of y = -20 for various times. That is, you want to find the area inside the box bounded by the y-axis, the x-axis, the line y = -20, and the line x = t where you pick various times t. You will need a rectangle to estimate this area, so click once on the rectangle near the upper right hand corner of the graph area to get a usable rectangle to drop out. Then, click and drag this rectangle to where you need it. Resize the rectangle to fit the area for the time you're looking at. Once the rectangle is fit fairly closely, click the "Calculate Area" button, which will display a number that is the value of the area at that time. That area is how much the car has slowed down in that amount of time.

To figure out what the car's speed is at that time, you need to subtract that area from the car's initial speed at time 0 seconds, which is given to be 88 ft/sec.

For example, if you want to find the area under y = -20 at time 1 second, you would move and resize the rectangle so that it covered 0 to 1 on the x-axis and 0 to -20 on the y-axis. When you click the "Calculate Area" button you should get near to 20. Since the area is below the x-axis, then the area is negative (i.e., -20). This means that the car's speed has slowed down by 20 ft/sec. So after 1 second the car's speed is 88 - 20 = 68 ft/sec.

After repeating this procedure for various times, you will have a set of points representing an estimate of the car's speed versus time: at time t = 0 the car's given speed is 88, at time t = 1 the car's speed as calculated above is 68. So, two of the points are (0, 88) and (1, 68).

Next: Follow the "Velocity Curve" link to the left.