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# What are position vs. time graphs?

See what we can learn from graphs that relate position and time.

## How are position vs. time graphs useful?

Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a conveniently small space.

## What does the vertical axis represent on a position graph?

The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.

Concept check: What is the position of the object at time t, equals, 5 seconds according to the graph above?

## What does the slope represent on a position graph?

The slope of a position graph represents the velocity of the object. So the value of the slope at a particular time represents the velocity of the object at that instant.
To see why, consider the slope of the position vs. time graph shown below:
The slope of this position graph is start text, s, l, o, p, e, end text, equals, start fraction, start text, r, i, s, e, end text, divided by, start text, r, u, n, end text, end fraction, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction.
This expression for slope is the same as the definition of velocity: v, equals, start fraction, delta, x, divided by, delta, t, end fraction, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction. So the slope of a position graph has to equal the velocity.
This is also true for a position graph where the slope is changing. For the example graph of position vs. time below, the red line shows you the slope at a particular time. Try sliding the dot below horizontally to see what the slope of the graph looks like for particular moments in time.

The slope of the curve between the times t, equals, 0, start text, space, s, end text and t, equals, 3, start text, space, s, end text is positive since the slope is directed upward. This means the velocity is positive and the object is moving in the positive direction.
The slope of the curve is negative between t, equals, 3, start text, space, s, end text and t, equals, 9, start text, space, s, end text since the slope is directed downward. This means that the velocity is negative and the object is moving in the negative direction.
At t, equals, 3, start text, space, s, end text, the slope is zero since the line representing the slope is horizontal. This means the velocity is zero and the object is momentarily at rest.
Concept check: What is the velocity of the object at t, equals, 9, start text, space, s, end text according to the graph above?
One more thing to keep in mind is that the slope of a position graph at a given moment in time gives you the instantaneous velocity at that moment in time. The average slope between two points in time will give you the average velocity between those two points in time. The instantaneous velocity does not have to equal the average velocity. However, if the slope is constant for a period of time (i.e., the graph is a straight line segment), then the instantaneous velocity will equal the average velocity between any two points on that line segment.

## What does the curvature on a position graph mean?

Look at the graph below. It looks curvy since it's not just made out of straight line segments. If a position graph is curved, the slope will be changing, which also means the velocity is changing. Changing velocity implies acceleration. So, curvature in a graph means the object is accelerating, changing velocity/slope.
On the graph below, try sliding the dot horizontally to watch the slope change. The first hump between 1, start text, space, s, end text and 5, start text, space, s, end text represents negative acceleration since the slope goes from positive to negative. For the second hump between 7, start text, space, s, end text and 11, start text, space, s, end text, the acceleration is positive since the slope goes from negative to positive.

Concept check: What is the acceleration of the object at t, equals, 6, start text, space, s, end text according to the graph above?
To summarize, if the curvature of the position graph looks like an upside down bowl, the acceleration will be negative. If the curvature looks like a right side up bowl, the acceleration will be positive. Here's a way to remember it: if your bowl is upside down all your food will fall out and that is negative. If your bowl is right side up, all your food will stay in it and that is positive.

## What do solved examples involving position vs. time graphs look like?

### Example 1: Hungry walrus

The motion of a hungry walrus walking back and forth horizontally looking for food is given by the graph below, which shows the horizontal position x as a function of time t.
What was the instantaneous velocity of the walrus at the following times: 2, start text, space, s, end text, 5, start text, space, s, end text, and 8, start text, space, s, end text?

#### Finding the velocity at $2\text{ s}$2, start text, space, s, end text:

We can find the velocity of the walrus at t, equals, 2, start text, space, s, end text by finding the slope of the graph at t, equals, 2, start text, space, s, end text:
start text, s, l, o, p, e, end text, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction, start text, left parenthesis, u, s, e, space, t, h, e, space, f, o, r, m, u, l, a, space, f, o, r, space, s, l, o, p, e, right parenthesis, end text
Now we will pick two points along the line we are considering that conveniently lie at a hashmark so we can determine the value of the graph at those points. We'll choose the points left parenthesis, 0, start text, space, s, end text, comma, 1, start text, space, m, end text, right parenthesis and left parenthesis, 4, start text, space, s, end text, comma, 3, start text, space, m, end text, right parenthesis, but we could pick any two points between 0, start text, space, s, end text and 4, start text, space, s, end text. We must plug in the later point in time as point 2, and the earlier point in time as point 1.
start text, s, l, o, p, e, end text, equals, start fraction, 3, start text, space, m, end text, minus, 1, start text, space, m, end text, divided by, 4, start text, space, s, end text, minus, 0, start text, space, s, end text, end fraction, start text, left parenthesis, P, i, c, k, space, t, w, o, space, p, o, i, n, t, s, space, a, n, d, space, p, l, u, g, space, t, h, e, space, x, space, v, a, l, u, e, s, space, i, n, t, o, space, t, h, e, space, n, u, m, e, r, a, t, o, r, space, a, n, d, space, t, h, e, space, t, space, v, a, l, u, e, s, space, i, n, t, o, space, t, h, e, space, d, e, n, o, m, i, n, a, t, o, r, point, right parenthesis, end text
start text, s, l, o, p, e, end text, equals, start fraction, 2, start text, space, m, end text, divided by, 4, start text, space, s, end text, end fraction, equals, start fraction, 1, divided by, 2, end fraction, start text, space, m, slash, s, end text, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, point, right parenthesis, end text
So, the velocity of the walrus at 2, start text, space, s, end text was 0, point, 5, start text, space, m, slash, s, end text.

#### Finding the velocity at $5\text{ s}$5, start text, space, s, end text:

To find the velocity at 5, start text, space, s, end text, we just have to note that the graph is horizontal there. Since the graph is horizontal, the slope is equal to zero, which means that the velocity of the walrus at 5, start text, space, s, end text was 0, start text, space, m, slash, s, end text.

#### Finding the velocity at $8\text{ s}$8, start text, space, s, end text:

start text, s, l, o, p, e, end text, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction, start text, left parenthesis, U, s, e, space, t, h, e, space, f, o, r, m, u, l, a, space, f, o, r, space, s, l, o, p, e, point, right parenthesis, end text
We'll pick the points at the beginning and end of the final line segment, which are left parenthesis, 6, start text, space, s, end text, comma, 3, start text, space, m, end text, right parenthesis and left parenthesis, 9, start text, space, s, end text, comma, 0, start text, space, m, end text, right parenthesis.
start text, s, l, o, p, e, end text, equals, start fraction, 0, start text, space, m, end text, minus, 3, start text, space, m, end text, divided by, 9, start text, space, s, end text, minus, 6, start text, space, s, end text, end fraction, start text, left parenthesis, P, i, c, k, space, t, w, o, space, p, o, i, n, t, s, space, a, n, d, space, p, l, u, g, space, t, h, e, space, x, space, v, a, l, u, e, s, space, i, n, t, o, space, t, h, e, space, n, u, m, e, r, a, t, o, r, space, a, n, d, space, t, h, e, space, t, space, v, a, l, u, e, s, space, i, n, t, o, space, t, h, e, space, d, e, n, o, m, i, n, a, t, o, r, point, right parenthesis, end text
start text, s, l, o, p, e, end text, equals, start fraction, minus, 3, start text, space, m, end text, divided by, 3, start text, space, s, end text, end fraction, equals, minus, 1, start text, space, m, slash, s, end text, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, point, right parenthesis, end text
So, the velocity of the walrus at 8, start text, space, s, end text was minus, 1, start text, space, m, slash, s, end text.

### Example 2: Happy bird

The motion of an extraordinarily jubilant bird flying straight up and down is given by the graph below, which shows the vertical position y as a function of time t. Answer the following questions about the motion of the bird.
What was the average velocity of the bird between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text?
What was the average speed of the bird between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text?

#### Finding the average velocity of the bird between $t=0\text{ s}$t, equals, 0, start text, space, s, end text and $t=10\text{ s}$t, equals, 10, start text, space, s, end text:

To find the average velocity between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text, we can find the average slope between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text. Visually, this would correspond to finding the slope of the line that connects the initial point and the final point on the graph.
start text, s, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction, start text, left parenthesis, U, s, e, space, t, h, e, space, f, o, r, m, u, l, a, space, f, o, r, space, s, l, o, p, e, point, right parenthesis, end text
The initial point would be left parenthesis, 0, start text, space, s, end text, comma, 7, start text, space, m, end text, right parenthesis, and the final point would be left parenthesis, 10, start text, space, s, end text, comma, 6, start text, space, m, end text, right parenthesis.
start text, s, l, o, p, e, end text, equals, start fraction, 6, start text, space, m, end text, minus, 7, start text, space, m, end text, divided by, 10, start text, space, s, end text, minus, 0, start text, space, s, end text, end fraction, start text, left parenthesis, P, i, c, k, space, t, h, e, space, f, i, n, a, l, space, a, n, d, space, i, n, i, t, i, a, l, space, p, o, i, n, t, s, space, o, f, space, t, h, e, space, t, i, m, e, space, i, n, t, e, r, v, a, l, comma, space, a, n, d, space, p, l, u, g, space, i, n, space, v, a, l, u, e, s, point, right parenthesis, end text
start text, s, l, o, p, e, end text, equals, start fraction, minus, 1, start text, space, m, end text, divided by, 10, start text, space, s, end text, end fraction, equals, minus, 0, point, 1, start text, space, m, slash, s, end text, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, point, right parenthesis, end text
So, the average velocity of the bird between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text was minus, 0, point, 1, start text, space, m, slash, s, end text.

#### Finding the average speed of the bird between $t=0\text{ s}$t, equals, 0, start text, space, s, end text and $t=10\text{ s}$t, equals, 10, start text, space, s, end text:

The definition of average speed is the distance traveled divided by the time. So, to find the distance traveled, we need to add the path lengths of each leg of the trip. Between t, equals, 0, start text, space, s, end text and t, equals, 2, point, 5, start text, space, s, end text, the bird moved 5, start text, space, m, end text down. Then, between t, equals, 2, point, 5, start text, space, s, end text and t, equals, 5, start text, space, s, end text, the bird did not move at all. And finally between t, equals, 5, start text, space, s, end text and t, equals, 10, start text, space, s, end text, the bird flew 4, start text, space, m, end text upward. Adding up all the path lengths gives us a total distance traveled of start text, d, i, s, t, a, n, c, e, end text, equals, 9, start text, space, m, end text.
Now we can divide by the time to get the average speed s, start subscript, a, v, g, end subscript:
s, start subscript, a, v, g, end subscript, equals, start fraction, start text, d, i, s, t, a, n, c, e, end text, divided by, delta, t, end fraction, start text, left parenthesis, U, s, e, space, f, o, r, m, u, l, a, space, f, o, r, space, a, v, e, r, a, g, e, space, s, p, e, e, d, point, right parenthesis, end text
s, start subscript, a, v, g, end subscript, equals, start fraction, 9, start text, space, m, end text, divided by, 10, start text, space, s, end text, end fraction, equals, 0, point, 9, start text, space, m, slash, s, end text, start text, left parenthesis, P, l, u, g, space, i, n, space, v, a, l, u, e, s, comma, space, t, h, e, n, space, c, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, !, right parenthesis, end text
So the average speed of the bird between t, equals, 0, start text, space, s, end text and t, equals, 10, start text, space, s, end text was 0, point, 9, start text, space, m, slash, s, end text.

## Want to join the conversation?

• are there ways to calculate the slope of a curved graph without using calculus •  It turns out to be possible for the conic sections: circles, parabolas, hyperbolas, and ellipses, but I think that's about it for the functions used by most people today. You could invent or define some curves by what you want their slopes to do, and before Newton came along, people played with these a lot -- osculating curves and evolutes and such. But Newton basically INVENTED CALCULUS precisely because he needed to calculate the slopes of curved graphs of given functions, and there was no way to do it.
• In the average velocity, why don't we simply calculate the instantaneous velocity when t=0 and when t=10, add them and divide them over 2 to get their average? It gave me a different result! what's wrong then? • it's because "average velocity" in physics is different from the "average of the initial and final velocity". This is admittedly confusing, but the definition of average velocity is displacement over time. The definition of the average of the velocities is the sum of the velocities divided by two. Like Andrew said, if the acceleration was constant then it turns out these two quantities will be equal. But if acceleration was not constant you can't assume they will give the same answer.
• What is the meaning of negative acceleration? Since an object cannot slow down/decelerate beyond zero.
The velocity is negative when object moves in the opposite direction(the negative direction) so is negative acceleration the acceleration when the object is moving in the opposite direction(the negative direction)? • I have a few questions:

1. Since velocity is "Speed with given direction", and the acceleration is negative when the slope is going down, why is the velocity constant when the slope is constant? (Refer to graph 4)

2. In example 2, "The motion of an extraordinarily jubilant bird flying straight up and down is given by the graph..." states that the bird flies STRAIGHT UP AND DOWN. But why is the slope of the graph down and then up? Is the sentence just an intro?

3. If the slope is going up, the acceleration remains at a constant rate and will not increase anymore unless the slope goes even higher, the same goes for the velocity, am I right? • 1. Velocity is the slope of position vs. time. If that slope is not changing, the velocity is constant. If the slope is negative and not changing, the velocity is a negative constant.
Acceleration is slope of velocity vs time.

2. Yes, it's an introduction.

3. Can't tell what slope you are referring to, so can't answer.
• For the Hungry Walrus question, what does -1 m/s velocity mean? At first I thought that it meant the walrus was going slower, but actually the walrus went faster, right? What does the -1 imply? He was going back in the direction he came from? • Yes, the (-) tells us that he is going back in the direction he came from. And yes, he is actually going faster.
At 2 s -> slope = 0.5 m/s.
At 5 s -> slope = 0 m/s.
At 8 s -> slope = -1 m/s.
At 8 s the MAGNITUDE or SIZE (aka number) for the velocity is the greatest from the three (since 1 > 0.5 > 1). Thus, he goes faster at the end.
As for the signs, we only have them to indicate direction, since VELOCITY is speed with direction. For example, if we were just calculating SPEED, which has no direction, we would not put the (-). However, since we were calculating VELOCITY, which has direction, we put the (-) because he went back in the direction he came.

Hope that helps. :)
• Why does the slope relate to the velocity and not the speed?     