# Relative Motion - Using relative velocities in one dimension

Author(s):
Larry Gladney and Dennis DeTurck

### A problem inspired by the driver's manual and the white Bronco

In the simple example in the preceding section, you have seen that the relative position of two objects moving in one dimension is simply the difference in their positions (measured in any reference system). Relative velocity is defined in the obvious way as the derivative of the relative position -- which is of course the difference of the velocities of the objects. Relative acceleration is defined the same way.

### Driver's Ed redux

It is a basic principle of physics -- called Gallilean relativity -- that the laws of physics are the same when measured with respect to two reference systems that are moving at constant velocity with respect to one another. This can be especially useful when solving motion problems if one of the objects in the problem is moving with constant velocity relative to the "external" frame. A good example of this is provided by the following "classic" problem. You should solve it twice -- once from the point of view of an external stationary observer, and again from the point of view of the truck. Perhaps you can even solve it a third time, from the point of view of the car. At the end you will find an animation from the truck's point of view that reveals the answer to the problem.

Problem 4. The diagram above shows the passing ability of an automobile at low speed. From the data supplied in the figure, calculate the acceleration of the automobile during the pass and the time required for the pass. Assume constant acceleration. After finding the acceleration and the time required for the pass, use Maple to plot the position of both the automobile and the truck as a function of time. Using the graph, find the time corresponding to the point at which the automobile just passes the truck (i.e. the back of the car just passes the front of the truck).

( Follow this link to see an animated visual hint, and the answer.) [The video part of this no longer works. Ed.]

Larry Gladney and Dennis DeTurck, "Relative Motion - Using relative velocities in one dimension," Convergence (November 2004)

## JOMA

Journal of Online Mathematics and its Applications