Average Velocity
How about the average velocity between t = 0 and t = 3? It’s actually easier to sort this out with a graph in front of us, because it’s easy to see the displacement at t = 0 and t = 3, and so that we don’t confuse displacement and distance.
Average Speed
Although the total displacement in the first three seconds is one centimeter to the right, the total distance traveled is two centimeters to the left, and then three centimeters to the right, for a grand total of five centimeters. Thus, the average speed is not the same as the average velocity of the ant. Once we’ve calculated the total distance traveled by the ant, though, calculating its average speed is not difficult:
Curved Position vs. Time Graphs
This is all well and good, but how do you calculate the velocity of a curved position vs. time graph? Well, the bad news is that you’d need calculus. The good news is that SAT II Physics doesn’t expect you to use calculus, so if you are given a curved position vs. time graph, you will only be asked qualitative questions and won’t be expected to make any calculations. A few points on the graph will probably be labeled, and you will have to identify which point has the greatest or least velocity. Remember, the point with the greatest slope has the greatest velocity, and the point with the least slope has the least velocity. The turning points of the graph, the tops of the “hills” and the bottoms of the “valleys” where the slope is zero, have zero velocity.
In this graph, for example, the velocity is zero at points A and C, greatest at point D, and smallest at point B. The velocity at point B is smallest because the slope at that point is negative. Because velocity is a vector quantity, the velocity at B would be a large negative number. However, the speed at B is greater even than the speed at D: speed is a scalar quantity, and so it is always positive. The slope at B is even steeper than at D, so the speed is greatest at B.
Velocity vs. Time Graphs
Velocity vs. time graphs are the most eloquent kind of graph we’ll be looking at here. They tell us very directly what the velocity of an object is at any given time, and they provide subtle means for determining both the position and acceleration of the same object over time. The “object” whose velocity is graphed below is our ever-industrious ant, a little later in the day.
We can learn two things about the ant’s velocity by a quick glance at the graph. First, we can tell exactly how fast it is going at any given time. For instance, we can see that, two seconds after it started to move, the ant is moving at 2 cm/s. Second, we can tell in which direction the ant is moving. From t = 0 to t = 4, the velocity is positive, meaning that the ant is moving to the right. From t = 4 to t = 7, the velocity is negative, meaning that the ant is moving to the left.
