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# Analyzing straight-line motion graphically

Learn how to analyze a particles motion given the graph of its position over time. Created by Sal Khan.

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• What does this have to do with calculus?
• Calculus is the mathematics of change. Velocity is a derivative of position, etc. It is a rate too. It is how much does the position change per unit of time.
• Can someone summarize the meaning of this graph for me? I know the connection between the functions: ds/dt is v(t) and dv/dt is a(t). However, I am confused with the positive and negative thing.

For example: When t > 3, the position decreases but why does the velocity slightly increase? If the velocity keeps increasing (for example v > 0), what will happen to the position?
• Well, let's make up a function and some characteristics in order to more easily understand this. It goes from 0 to -1 during time (0,1), and then -1 to -1.5 during time (1,2). So, during the first time interval, the velocity is -1. In the second time interval, the position is still decreasing but at a slower rate, so the velocity is now -0.5. It is harder to see in this video, but you can see that the position is decreasing at a slower and slower rate as time goes by. That is why the velocity then becomes less and less negative, and the acceleration actually becomes slightly positive.
• At , why isn't 1 included but 0 is?
• We include 0 because the graph indicates positive velocity at that point, which means the particle is moving to the right. We exclude 1 because the graph indicates velocity is zero at that point, so the particle can't be moving to the right (it's not moving at all).
• I understand how calculus is used to analyze particle motion here, but, in the real world, how would you get the position function in the first place?
• Why does the acceleration cross the x axis at 2 instead or 1? Surely the acceleration would be 0 at t=1?
• At t=1 velocity (red line) keep changing, So the acceleration cannot be 0 at this moment. It seems that velocity get to a constant level (no changes) closer to t=2.
(1 vote)
• Hello, how can a velocity (also acceleration) be negative (at some points)? thanks Irena
(1 vote)
• A negative velocity means that you are going backward. A negative acceleration means you are slowing down.
• For the last part of the question, finding total distance traveled, Sal used the position function. While we don't have the necessary equations, would he have gotten the same answer if found the absolute value of the area under the velocity curve over the same time period?
(1 vote)
• Yes. Because the velocity is the derivative of the position function, the position function is the "antiderivative" of the velocity function, also known as the indefinite integral of the function. This indefinite integral can be used to find the area underneath the curve of the velocity curve and will give you the total distance traveled.
• In what kind of situation do these values actually come from. Literally particles such as electrons and stuff moving? Or is this all made up for the sake of the question?
• Well, these ideas can be translated to any time of movement or graph that represents movement. It can be represented about our movement in a car for instance. However, this isn't always 'fun' in mathematics so we use particles.

But it is very, very important in physics and science in general.
• Sal,in the 2nd question for the time interval, why is only the zero in square brackets and not one?