# Position vs. Time Graphs

#### Beanie

1. Homework Statement
Three objects can only move along a straight, level path. The graphs below show the po- sition d of each of the objects plotted as a function of time t.

The magnitude of the velocity ∥⃗v∥ of the object increases in which of the cases? (view image/attached file)

2. Homework Equations

3. The Attempt at a Solution
I thought the correct answer was graphs 1 and 3 because the distance in graphs 1 and 3 changes over time. Therefore, the velocity must be increasing over time. However this answer is not correct.

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#### brainpushups

I don't see an attempt. What do you think?

#### Beanie

I don't see an attempt. What do you think?
I initially thought that the answer was graphs I and II

#### brainpushups

I initially thought that the answer was graphs I and II
Why?

Edit:
Sorry about the attachment; accidentally hit paste and there was an image in my clipboard. Couldn't figure out how to delete the attachment.

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#### Beanie

Why?

I thought the correct answer was graphs 1 and 2 because the distance in graphs 1 and 2 changes over time. Therefore, the velocity must be increasing over time.

#### brainpushups

Actually the distance does NOT change in graph II. If it did then there would need to be some variation in the vertical direction (y direction of the axes).

One quick point: The question asks about changing speed, not changing distance.

Let me ask you this: is it possible to be in motion without changing your speed?

Edit: for clarity

#### Beanie

Sorry, I meant to write graphs 1 and 3 instead of graphs 1 and 2. I was thinking along the lines that if distance is changing, velocity must be increasing. But I realize now that that is not necessarily true because the object could be changing positions with a constant velocity. So how can I know whether velocity is increasing or not by looking at a distance vs. time graph?

#### brainpushups

Two questions:

Suppose you are moving 2 meters per second. What would the distance vs time graph of this motion look like? If you are unsure, then calculate the distance for a couple of different times (1 second, 2 seconds, etc) and make a chart then sketch the graph.

What does the slope of the line you just sketched represent?

#### Beanie

In the situation you gave, the graph would look like an upwards slope, like graph 1. The slope of the line represents that as time goes on the distance travelled increases.

#### brainpushups

Indeed. The graph is a straight line with positive slope. And while you are not incorrect about how you stated the meaning of slope there is a more eloquent way of expressing what the slope represents. Check out the units of the slope - you should notice that they are units of velocity (rise/run = m/s in the example I gave you). So... the slope of the distance vs. time graph is equal to the velocity.

If the slope of the curve on a distance vs. time graph represents the velocity and you are asked to identify cases in which the velocity is changing then you need to find the graph for which the slope is not constant. Does that make sense?

#### Beanie

Indeed. The graph is a straight line with positive slope. And while you are not incorrect about how you stated the meaning of slope there is a more eloquent way of expressing what the slope represents. Check out the units of the slope - you should notice that they are units of velocity (rise/run = m/s in the example I gave you). So... the slope of the distance vs. time graph is equal to the velocity.

If the slope of the curve on a distance vs. time graph represents the velocity and you are asked to identify cases in which the velocity is changing then you need to find the graph for which the slope is not constant. Does that make sense?
Yes, thank you. This makes much more sense.

So just to recap, the slope of a distance vs. time graph is m/s, and therefore velocity. So when it asks for a graph where the velocity is increasing, I must find a graph where the velocity is not constant, meaning the slope is not constant.

So the correct answer is graph III?

#### brainpushups

So just to recap, the slope of a distance vs. time graph is m/s, and therefore velocity
The slope of the distance vs. time graph will have units of distance/time. For the particular example I gave the units were m/s, but they could just as easily be km/h.

So when it asks for a graph where the velocity is increasing, I must find a graph where the velocity is not constant, meaning the slope is not constant.
You are asked to identify a graph for which the object's speed is increasing so you need to find a graph of distance vs. time for which the slope is increasing.

So the correct answer is graph III?
Yes.

#### Beanie

The slope of the distance vs. time graph will have units of distance/time. For the particular example I gave the units were m/s, but they could just as easily be km/h.

You are asked to identify a graph for which the object's speed is increasing so you need to find a graph of distance vs. time for which the slope is increasing.

Yes.
Thank you!