Video: Velocity Graphs

The diagram shows a graph of the variation of the position 𝑥 of an object with time 𝑡. What is the velocity of the object in the time interval 𝑡 > 0 s to 𝑡 = 0.5 s? What is the velocity of the object in the time interval 𝑡 > 0.5 s to 𝑡 = 1.0 s? What is the velocity of the object in the time interval 𝑡 > 1.0 s to 𝑡 = 2.0 s?

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Video Transcript

The diagram shows a graph of the variation of the position 𝑥 of an object with time 𝑡. What is the velocity of the object in the time interval 𝑡 greater than zero seconds to 𝑡 equal 0.5 seconds? What is the velocity of the object in the time interval 𝑡 greater than 0.5 seconds to 𝑡 equals 1.0 seconds? What is the velocity of the object in the time interval 𝑡 greater than 1.0 seconds to 𝑡 equals 2.0 seconds?

In this exercise, we want to solve for three average velocities. We’ll call them 𝑣 one, 𝑣 two, and 𝑣 three. These average velocities are based on our position versus time graph, where 𝑣 one applies to the first leg of the journey, 𝑣 two applies to the second leg, and 𝑣 three applies to the third and final leg of this movement. To get started, we can recall that average velocity 𝑣 sub avg is equal to displacement divided by time 𝑡. Let’s apply this relationship to solve first for 𝑣 sub one.

𝑣 sub one is equal to the position of our object 𝑝 when time equals 0.5 seconds minus our object’s position when time equals zero seconds. And this difference in position, called the displacement, is then divided by the time it takes to move that distance. Looking at our diagram, we can mark out the position of the object at 𝑡 equals 0.5 seconds and 𝑡 equals zero seconds. And if we drop a vertical line down from our second point, we see that the time interval Δ𝑡 for all this to happen is equal to 0.5 seconds. This means that 𝑣 sub one is 1.0 meters per second. That’s the average velocity of the object over this initial time interval.

Next, we move on to calculating 𝑣 sub two. 𝑣 sub two is the average velocity of the object between time equals 1.0 seconds and 0.5 seconds. Looking at the diagram of the object’s position versus time, we see that its position at both these time values is the same. It’s 0.5 meters. That means 𝑣 sub two is 0.0 meters per second. That’s the average velocity of our object over a time interval where its position doesn’t change.

Finally, we move on to calculating 𝑣 three, the average velocity of our object over the third time interval from time 𝑡 equals 2.0 seconds to 1.0 seconds. Looking on our diagram, we see that at 2.0 seconds our object has a position of 0.0 meters and at 1.0 seconds it has a position of 0.5 meters. Calculating this fraction, we find 𝑣 sub three is negative 0.50 meters per second. That’s the object’s average velocity on the last leg of its movement.

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