Since you are not allowed to use calculators, SAT II Physics places a heavy emphasis on qualitative problems. A common way of testing kinematics qualitatively is to present you with a graph plotting position vs. time, velocity vs. time, or acceleration vs. time and to ask you questions about the motion of the object represented by the graph. Because SAT II Physics is entirely made up of multiple-choice questions, you won’t need to know how to draw graphs; you’ll just have to interpret the data presented in them.
Knowing how to read such graphs quickly and accurately will not only help you solve problems of this sort, it will also help you visualize the often-abstract realm of kinematic equations. In the examples that follow, we will examine the movement of an ant running back and forth along a line.
Position vs. Time Graphs
Position vs. time graphs give you an easy and obvious way of determining an object’s displacement at any given time, and a subtler way of determining that object’s velocity at any given time. Let’s put these concepts into practice by looking at the following graph charting the movements of our friendly ant.
Any point on this graph gives us the position of the ant at a particular moment in time. For instance, the point at (2,–2) tells us that, two seconds after it started moving, the ant was two centimeters to the left of its starting position, and the point at (3,1) tells us that, three seconds after it started moving, the ant is one centimeter to the right of its starting position.
Let’s read what the graph can tell us about the ant’s movements. For the first two seconds, the ant is moving to the left. Then, in the next second, it reverses its direction and moves quickly to
y = 1. The ant then stays still at
y = 1 for three seconds before it turns left again and moves back to where it started. Note how concisely the graph displays all this
information.
Calculating Velocity
We know the ant’s displacement, and we know how long it takes to move from place to place. Armed with this
information, we should also be able to determine the ant’s velocity, since velocity measures the rate of change of displacement over time. If displacement is given here by the vector
y, then the velocity of the ant is
If you recall, the slope of a graph is a measure of rise over run; that is, the amount of change in the
y direction divided by the amount of change in the
x direction. In our graph,
is the change in the
y direction and
is the change in the
x direction, so
v is a measure of the slope of the graph.
For any position vs. time graph, the velocity at time t
is equal to the slope of the line at t
. In a graph made up of straight lines, like the one above, we can easily calculate the slope at each point on the graph, and hence know the instantaneous velocity at any given time.
We can tell that the ant has a velocity of zero from t = 3 to t = 6, because the slope of the line at these points is zero. We can also tell that the ant is cruising along at the fastest speed between t = 2 and t = 3, because the position vs. time graph is steepest between these points. Calculating the ant’s average velocity during this time interval is a simple matter of dividing rise by run, as we’ve learned in math class.
