Graphs of Motion; Velocity

Motion with constant velocity

m / Motion with constant velocity.

In example m, an object is moving at constant speed in one direction. We can tell this because every two seconds, its position changes by five meters. In algebra notation, we'd say that the graph of x vs. t shows the same change in position, Δx = 5.0 m, over each interval of Δt = 2.0 s. The object's velocity or speed is obtained by calculating v = Δx/Δt = (5.0 m)/(2.0 s) = 2.5 m/s. In graphical terms, the velocity can be interpreted as the slope of the line. Since the graph is a straight line, it wouldn't have mattered if we'd taken a longer time interval and calculated v = Δx/Δt = (10.0 m)/(4.0 s). The answer would still have been the same, 2.5 m/s.

Note that when we divide a number that has units of meters by another number that has units of seconds, we get units of meters per second, which can be written m/s. This is another case where we treat units as if they were algebra symbols, even though they're not.

n / Motion that decreases x is represented with negative values of Δx and v.

In example n, the object is moving in the opposite direction: as time progresses, its x coordinate decreases. Recalling the definition of the Δ notation as "after minus before," we find that Δt is still positive, but Δx must be negative. The slope of the line is therefore negative, and we say that the object has a negative velocity, v = Δx/Δt = (-5.0 m)/(2.0 s) = -2.5 m/s. We've already seen that the plus and minus signs of Δx values have the interpretation of telling us which direction the object moved. Since Δt is always positive, dividing by Δt doesn't change the plus or minus sign, and the plus and minus signs of velocities are to be interpreted in the same way. In graphical terms, a positive slope characterizes a line that goes up as we go to the right, and a negative slope tells us that the line went down as we went to the right.

→ Solved problem: light-years page 94, problem 4

Motion with changing velocity

Now what about a graph like figure o? This might be a graph of a car's motion as the driver cruises down the freeway, then slows down to look at a car crash by the side of the road, and then speeds up again, disappointed that there is nothing dramatic going on such as flames or babies trapped in their car seats. (Note that we are still talking about one-dimensional motion. Just because the graph is curvy doesn't mean that the car's path is curvy. The graph is not like a map, and the horizontal direction of the graph represents the passing of time, not distance.)

o / Motion with changing velocity.

Example o is similar to example m in that the object moves a total of 25.0 m in a period of 10.0 s, but it is no longer true that it makes the same amount of progress every second. There is no way to characterize the entire graph by a certain velocity or slope, because the velocity is different at every moment. It would be incorrect to say that because the car covered 25.0 m in 10.0 s, its velocity was 2.5 m/s. It moved faster than that at the beginning and end, but slower in the middle. There may have been certain instants at which the car was indeed going 2.5 m/s, but the speedometer swept past that value without "sticking," just as it swung through various other values of speed. (I definitely want my next car to have a speedometer calibrated in m/s and showing both negative and positive values.)

p / The velocity at any given moment is defined as the slope of the tangent line through the relevant point on the graph.

q / Example: finding the velocity at the point indicated with the dot.

r / Reversing the direction of motion.

We assume that our speedometer tells us what is happening to the speed of our car at every instant, but how can we define speed mathematically in a case like this? We can't just define it as the slope of the curvy graph, because a curve doesn't have a single well-defined slope as does a line. A mathematical definition that corresponded to the speedometer reading would have to be one that attached a different velocity value to a single point on the curve, i.e., a single instant in time, rather than to the entire graph. If we wish to define the speed at one instant such as the one marked with a dot, the best way to proceed is illustrated in p, where we have drawn the line through that point called the tangent line, the line that "hugs the curve." We can then adopt the following definition of velocity:

One interpretation of this definition is that the velocity tells us how many meters the object would have traveled in one second, if it had continued moving at the same speed for at least one second. To some people the graphical nature of this definition seems "inaccurate" or "not mathematical." The equation by itself, however, is only valid if the velocity is constant, and so cannot serve as a general definition.

The slope of the tangent line.

Conventions about graphing

The placement of t on the horizontal axis and x on the upright axis may seem like an arbitrary convention, or may even have disturbed you, since your algebra teacher always told you that x goes on the horizontal axis and y goes on the upright axis. There is a reason for doing it this way, however. In example q, we have an object that reverses its direction of motion twice. It can only be in one place at any given time, but there can be more than one time when it is at a given place. For instance, this object passed through x = 17 m on three separate occasions, but there is no way it could have been in more than one place at t = 5.0 s. Resurrecting some terminology you learned in your trigonometry course, we say that x is a function of t, but t is not a function of x. In situations such as this, there is a useful convention that the graph should be oriented so that any vertical line passes through the curve at only one point. Putting the x axis across the page and t upright would have violated this convention. To people who are used to interpreting graphs, a graph that violates this convention is as annoying as fingernails scratching on a chalkboard. We say that this is a graph of "x versus t." If the axes were the other way around, it would be a graph of "t versus x." I remember the "versus" terminology by visualizing the labels on the x and t axes and remembering that when you read, you go from left to right and from top to bottom.

Discussion Questions

A	Park is running slowly in gym class, but then he notices Jenna watching him, so he speeds up to try to impress her. Which of the graphs could represent his motion?
B	The figure shows a sequence of positions for two racing tractors. Compare the tractors' velocities as the race progresses. When do they have the same velocity? [Based on a question by Lillian McDermott.]
C	If an object had a straight-line motion graph with Δx=0 and Δt ≠ 0, what would be true about its velocity? What would this look like on a graph? What about Δt=0 and Δx ≠ 0?
D	If an object has a wavy motion graph like the one in example (e) on the previous page, which are the points at which the object reverses its direction? What is true about the object's velocity at these points?
E	Discuss anything unusual about the following three graphs.
F	I have been using the term "velocity" and avoiding the more common English word "speed," because introductory physics texts typically define them to mean different things. They use the word "speed," and the symbol "s" to mean the absolute value of the velocity, s = |v|. Although I've chosen not to emphasize this distinction in technical vocabulary, there are clearly two different concepts here. Can you think of an example of a graph of x-versus-t in which the object has constant speed, but not constant velocity?
G	For the graph shown in the figure, describe how the object's velocity changes.
H	Two physicists duck out of a boring scientific conference to go get beer. On the way to the bar, they witness an accident in which a pedestrian is injured by a hit-and-run driver. A criminal trial results, and they must testify. In her testimony, Dr. TransverzWaive says, "The car was moving along pretty fast, I'd say the velocity was +40 mi/hr. They saw the old lady too late, and even though they slammed on the brakes they still hit her before they stopped. Then they made a U turn and headed off at a velocity of about -20 mi/hr, I'd say." Dr. Longitud N.L. Vibrasheun says, "He was really going too fast, maybe his velocity was -35 or -40 mi/hr. After he hit Mrs. Hapless, he turned around and left at a velocity of, oh, I'd guess maybe +20 or +25 mi/hr." Is their testimony contradictory? Explain.