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# What are position vs. time graphs?

Learn how to interpret the motion of an object represented on a position vs. time graph.
A position vs. time graph represents the position of an object as a function of time.
For example, if you read the $y$-value of the graph below at a particular time (in seconds), you will get the position of the object (in meters) at that instant.
Try sliding the dot horizontally on the graph below to see the position at different times.
Concept check: According to the graph above, what is the object's position at time

## What does the slope of a position vs. time graph represent?

The slope of a position vs. time graph represents the velocity of the object.
To see why, consider the graph below, which represents the position, $x,$ of an object moving in one dimension as a function of time, $t$:
The slope of the graph is:
$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{{\stackrel{\to }{x}}_{2}-{\stackrel{\to }{x}}_{1}}{{t}_{2}-{t}_{1}}=\frac{\mathrm{\Delta }\stackrel{\to }{x}}{\mathrm{\Delta }t}={\stackrel{\to }{v}}_{avg}$
The slope of a position vs. time graph represents the change in the object's position divided by the time taken for that change. That's average velocity!
The graph above is a straight line—it has constant slope, so we know the object moved with constant velocity. Therefore the average velocity is the same as the velocity at any given time. But what if an object does NOT move with constant velocity? In that case, we should expect the slope of its position vs. time graph to change.
Consider the example below. We can tell right away that this object's velocity changed (it accelerated) over the time interval shown, since the slope changes. To know what the velocity was at a given instant—the instantaneous velocity—we need the slope at that instant. In other words, we need the slope of the line tangent to the curve at that time.
Try sliding the dot horizontally on the graph below to see the slope of the tangent line at different times.
Concept check: At what time(s) did the object have an instantaneous velocity of $v=0\text{?}$
Concept check: From to was the object moving in the $+$ or $-$ direction? Did it speed up, slow down, or both?
Concept check: From to was the object moving in the $+$ or $-$ direction? Did it speed up, slow down, or both?

## Average velocity

Above, we considered the instantaneous velocity of an object, equal to the slope of its position vs. time graph at an instant. But sometimes, we want to know the average velocity of an object over an interval.
On a position vs. time graph, the average velocity for an interval is equal to the slope of the line connecting the endpoints of the interval.
Consider the position vs. time graph below.
Concept check: What was the average velocity of the object from to
Concept check: What was the average velocity of the object from to
Concept check: What was the average velocity of the object from to

## Common mistakes and misconceptions

• The position vs. time graphs above represent motion in one dimension. The object did NOT trace out the shape on the graph. For example, if a graph goes up and then down, that doesn't necessarily mean the object went up and then down. The object could have gone forward and then backward. Do not think of the shape of the graph as the object's path. Instead, it is a record of the object's position as a function of time.
• If the y-value of the graph is positive (the line is above the x-axis), the object was located at a positive position at that time. If the slope of the graph is positive (tilted up), the object had positive velocity at that time. At any time the position and velocity may both be positive, both be negative, or have different signs. It's very easy to confuse them. Remember to only consider slope when determining velocity.
• The average velocity over an interval may or may not equal the instantaneous velocity at all times throughout that interval.

## 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)?
• Yes, negative acceleration would be acceleration in the negative direction. Acceleration is a vector quantity.
• 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?
• As displacement is a vector quantity, the slope of dispacement-time graph should be velocity because velocity is a vector quantity as well.
(Vector quantities have a direction and magnitude)
(1 vote)
• In example two, wouldn't the variable x in the formula change to y since it is measuring vertical position?