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Velocity Time Plots

A rather useful tool for understanding kinematics for people not well versed with calculus is a velocity-time plot.

Below is a graph of an object's position, velocity, and acceleration with respect to time. For this example, I used constant acceleration because it's easier to draw, but there is another example with a more complicated acceleration below that.


Usually, you will be given the velocity-time plot first (middle) and asked to determine the information for acceleration using rise-over-run, or the slope of the line. The position plot is the "area under the curve" of the velocity plot (aka an integral).


For this plot, the velocity has straight lines so the area can be determined with triangles, squares, and trapazoids. The area enclosed between the curve and the x-axis (zero velocity) is the change in x position, so the first triangle with an area of 2 (m/s * s) gives an increase of 2 m. The next 2 triangles enclose a negative velocity, so the particle moves backward.

The last thing to note here is the integral of the velocity only shows the change in a position not necessarily the absolute position in your preferred coordinate system, for example, imagine looking at someone waking back and forth through a telescope and counting their steps. From their steps, you can find how their position changes from when you started watching, but if you don't know how far they started from you, you can't figure out where they end with respect to you. to convert to a "preferred coordinate system" add a constant to the position (not starting at zero). This shifts the whole line up or down, but does not affect the slope of that line.

The same process as above can be repeated on non-constant accelerations, but it's a lot messier when done by hand.

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