Mr. Rogers' AP Physics C: Kinematics Problems
Two-Part Kinematics Problem: Starting from rest Martha accelerates her motor cycle forward at 5 m/s2 for ten seconds then accelerates backwards at 2 m/s2 for ten seconds. How fast is she going at the end of the 20 second time span described above? How far has she traveled?  Sketch the a vs.t, v vs. t, x vs. t graphs.

Solution

Define forward as positive.  While the acceleration is not constant for the entire 20 seconds, clearly, it is constant during the first 10 seconds with one value and during the second 10 seconds with another value. The a vs.t, v vs. t, x vs. t are as shown at bottom right.

 First find the velocity at the end of part 1: v1 = at + vo = (5 m/s2)(10 s) + 0 = 50 m/s Next find the displacement in part 1 x1 = 1/2at2 + vo t + xo = 1/2 (5 m/s2) (10 s)2 + 0 + 0 = 250 m Now find the final velocity of part 2 v2 = at + v1 = (-2m/s2)(10 s) + 50 v2 = 30 m/s is the final velocity Finally find the total distance traved x2 = 1/2at2 + vo t + xo = 1/2 (-2 m/s2) (10 s)2 + (50 m/s) (10 s) + (250 m) = (- 100 + 500 +250) m x2 = 650 m is the total distance traveled

Conclusion and Significance

When splitting a problem into multiple parts, the end conditions of the previous part become the initial conditions of the current part.

Metacognition Questions:

Is the acceleration constant in different parts of the problem? With constant acceleration there are essentially just two equations that can be written:
 v = at + vo x = 1/2at2 + vo t + xo

These equations are only valid for constant acceleration, but sometimes a problem can be split into separate parts in which the acceleration is constant.