Determining Slope for Position vs. Time Graphs

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  • 0:02 Building Physics
  • 0:31 Slope and Velocity
  • 2:30 Avg. Velocity Example Problem
  • 4:33 Speed Example Problem
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Angela Hartsock

Angela has taught college Microbiology and has a doctoral degree in Microbiology.

Simply looking at a position vs. time graph can tell you a lot about straight line motion, but doing a few basic calculations can tell you even more. In this lesson, we will learn how to use the slope of the line to determine average velocity.

Building Physics

Learning physics is a lot like playing with blocks. Before you can build a tower, you need to make a solid base. Before you describe the motion of the planets, you need to start with something simpler and describe the motion of a car driving along a straight road.

In another lesson, we discussed graphing the position versus the time for an object in motion. Now it's time to add another layer of knowledge and learn how this graph can be used to calculate the average velocity of your moving object.

The Relationship Between Slope and Velocity

In algebra class, probably about the time you were introduced to the x-y scatter graph, you not only learned how to plot data points, but you learned how to calculate the slope of the line as well. The slope is defined as the change in y divided by the change in x. Let's write this a slightly different way.

Slope = Δy / Δx

Remember, the triangular symbol delta (Δ) means 'change in.'

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But let's look at our graph. On a position vs. time graph, y is position, which we can also call displacement (represented by the letter s). Displacement has the unit meters. x is time (represented by the letter t) and has the unit seconds. Let's substitute these values into our equation:

Slope = Δs / Δt

Does this equation look familiar? It should. This is the equation for calculating average velocity.

Average Velocity = Δs / Δt

Need a bit more convincing? The units for displacement are meters (m). The units for time are seconds (s). So, displacement divided by time gives an answer with meters per second (m/s) as units, the exact units attached to average velocity.

This was a very long-winded attempt on my part to tell you that the slope of a position vs. time graph gives you the velocity of the object in motion. Hopefully, by laying it out this way, it'll stick in your brain.

There is one quick point to be made here: velocity is a vector quantity. This means that there must be a directional component. Since we are only graphing objects traveling in a straight line, the only two directions we need to be concerned with are forwards and backwards. If the velocity has a positive sign, the object must be moving forwards. If the velocity has a negative sign, the object is moving backwards.

Average Velocity Example Problem

Let's look at a position vs. time graph so we can practice calculating the average velocity. First, let's determine the average velocity between 0 and 3 seconds.

Average Velocity = Δs / Δt

The easy variable to calculate is Δt. The question asks for the change in displacement between 0 and 3 seconds, so 3 seconds - 0 seconds = a change in time of 3 seconds. Now, simply follow the 3 second grid line up to where it intersects the blue graph line.

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Then, once you hit the intersection, if you read across to the position axis, you can see that this occurs at the 30 meter mark. Since the object started at the 0 meter mark, the change in position is 30 meters - 0 meters, which equals 30 meters. Now simply plug these values into the equation:

Average Velocity = Δs / Δt = 30 m / 3 s = 10 m/s

The average velocity over the first 3 seconds of the graph is 10 m/s.

We can also determine the average velocity over the entire portion of the graph. The last point on our graph is at -15 meters and 17 seconds. Remember, this is a vector quantity, so the total distance traveled is not important, only the displacement from the starting position, which is 0 meters at 0 seconds in this problem.

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