How to analyze graphs that relate velocity and time to acceleration and displacement.
What does the vertical axis represent on a velocity graph?
The vertical axis represents the velocity of the object. This probably sounds obvious, but be forewarned—velocity graphs are notoriously difficult to interpret. People get so used to finding velocity by determining the slope—as would be done with a position graph—they forget that for velocity graphs the value of the vertical axis is giving the velocity.
Try sliding the dot horizontally on the example graph below to choose different times and see how the velocity changes.
Concept check: What is the velocity of the object at time
The velocity is
What does the slope represent on a velocity graph?
The slope of a velocity graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
The slope of a velocity graph will be given by the following formula:
Since
This means that when the slope is steep, the object will be changing velocity rapidly. When the slope is shallow, the object will not be changing its velocity as rapidly. This also means that if the slope is negative—directed downwards—the acceleration will be negative, and if the slope is positive—directed upwards—the acceleration will be positive.
Try sliding the dot horizontally on the example velocity graph below to see what the slope looks like for particular moments in time.
The slope of the curve is positive between the times
The slope of the curve is negative between
At
Concept check: Is the object whose motion is described by the graph above speeding up or slowing down at time
The object is slowing down at
One way to see this is by dragging the slider to
Another way to see this is by recognizing that at
What does the area under a velocity graph represent?
The area under a velocity graph represents the displacement of the object. To see why, consider the following graph of motion that shows an object maintaining a constant velocity of 6 meters per second for a time of 5 seconds.
To find the displacement during this time interval, we could use this formula
which gives a displacement of
Now we're going to show that this was equivalent to finding the area under the curve. Consider the rectangle of area made by the graph as seen below.
The area of this rectangle can be found by multiplying height of the rectangle, 6 m/s, times its width, 5 s, which would give
This is the same answer we got before for the displacement. The area under a velocity curve, regardless of the shape, will equal the displacement during that time interval.
For an arbitrary shape, you can consider breaking the area into rectangles of very small width as seen in the graph below. The displacement during each small time interval would be given by
You would still find the area from the curve to the time axis. But since the area lies beneath the time axis, the displacement would be equal to negative the area. This makes sense since the velocity would be negative during this time which would lead to a negative displacement.
For this reason, many people think of areas that lie underneath the time axis as "negative areas under the curve" so that they remember to add the negative sign. Strictly speaking this is nonsense, since area is zero or positive by definition, but that doesn't stop people from using "negative area under the curve" as a useful mnemonic device.
What do solved examples involving velocity vs. time graphs look like?
Example 1: Windsurfing speed change
A windsurfer is traveling along a straight line, and her motion is given by the velocity graph below.
Select all of the following statements that are true about the speed and acceleration of the windsurfer.
(A) Speed is increasing.
(B) Acceleration is increasing.
(C) Speed is decreasing.
(D) Acceleration is decreasing.
Options A, speed increasing, and D, acceleration decreasing, are both true.
The slope of a velocity graph is the acceleration. Since the slope of the curve is decreasing and becoming less steep this means that the acceleration is also decreasing.
It might seem counterintuitive, but the windsurfer is speeding up for this entire graph. The value of the graph, which represents the velocity, is increasing for the entire motion shown, but the amount of increase per second is getting smaller. For the first 4.5 seconds, the speed increased from 0 m/s to about 5 m/s, but for the second 4.5 seconds, the speed increased from 5 m/s to only about 7 m/s.
Example 2: Go-kart acceleration
The motion of a go-kart is shown by the velocity vs. time graph below.
A. What was the acceleration of the go-kart at time
B. What was the displacement of the go-kart between
A. Finding the acceleration of the go-kart at
We can find the acceleration at
For our two points, we'll choose the start—
B. Finding the displacement of the go-kart between
We can find the displacement of the go-kart by finding the area under the velocity graph. The graph can be thought of as being a rectangle (between
The area of the rectangle is found by
The area of the triangle is found by
Adding these two areas together gives the total displacement.
