# Relative Motion - A Simple Example

Author(s):

### One-dimensional projectile motion and warmup exercises

Equations are used to describe the motion of objects. Usually, the independent variable in these equations is t, for time. Depending on whether the motion is taking place along a line (one dimension), in a plane (two dimensions) or in space (three dimensions), we use one, two or three functions to specify the position of the object at any time.

### A simple example

An object that is projected straight up from the surface of the earth and then is subject only to gravity moves within a straight vertical line. Its motion can therefore be described by one function, y(t), which gives the height of the object above the earth for any time t. If we decide to measure time in seconds and height in feet, and if the object is projected upward at time = 0 with a velocity of 400 feet per second, then the function that gives the height of the object at time t is

y(t) = 400t - 16t2

Actually, this function only works for t between 0 and 25 seconds -- because 25 seconds after the object is projected upward, it hits the ground [ y(25) = 0].

Now suppose there are two objects. The first is projected upward when time t = 0 with initial velocity 400 feet per second -- so its height at time t is given by the function y(t) described above. The second is kept on the ground until time t = 5 seconds, and then it is projected upward with initial velocity 320 feet per second.

Problem 1: What function describes the motion of the second object for 0 < t < 25 seconds? Note that the function must be defined in pieces, since its value is zero for 0 < t < 5, and then it is given by some other formula for t > 5.

Problem 2: When does the second object hit the ground?

You can explore the motion of the two objects, both from an "external" point of view and from the point of view of one or the other of the objects in this Maple worksheet.

Problem 3: In the worksheet, the motion is viewed from the first object. You should also view the action from the point of view of the second object. How are the two related?

Larry Gladney and Dennis DeTurck, "Relative Motion - A Simple Example," Loci (November 2004)

## JOMA

Journal of Online Mathematics and its Applications