Introduction to Physics Exam Solutions

Recall that the basic method of finding the displacement of an object is to determine the area underneath a plot of v vs. t for the object. Thus, to find the displacement of the object in this plot, just find the area of the shaded region shown below between 5 seconds and 10 seconds:

There are approximately 58.5 blocks in the shaded region. If the time axis is in units of seconds and the velocity axis is in units of meters per second, this makes the units of the area just meters, which would be appropriate for displacement. Thus the displacement is 58.5 m.

Acceleration is just the rate of change of the velocity, or more specifically, the slope of the v vs. t plot. The slope of the plot shown above at the point where 2 seconds have passed is shown below:

Since the slope of the line at this point is positive, the acceleration at that point is also positive.

The point on the plot where 12 seconds have passed is shown below:

Since the slope of the plot at that point is negative, the acceleration is also negative.

The point at which 8 seconds have passed is shown below:

Since the plot is just about horizontal at that location, the slope is close to zero. Again, since the slope gives the acceleration, the acceleration at that point must also be zero.

While the general techniques of area and slope may be applied to any plot, the equations we developed in class (using these techniques) started with the assumption that the acceleration is constant. It should be obvious from the answers to (b) and (c) that the acceleration in this example is not constant. Since the acceleration is not constant, the equations we developed in class do not apply to this example.