Linear Motion

Using the applet in the previous section, you have created an equation for velocity as a function of acceleration and time. That equation assumed that the initial velocity was zero; to solve the problems which we are about to undertake, you will need the more general equation formed by adding the initial velocity to the right hand side.

Our ultimate goal is to determine the extent of your injuries following your fall from the roof. The first step is to understand the fall itself: how long did it take you to fall, and what was your velocity just before you hit the ground. In the applet below, you will be given the height from which you fell. From our previous discussion, you know the initial velocity and the acceleration. Your final position is at y = 0 (the ground), so you should be able to solve the following system of two equations:

position as a function of the initial position, initial velocity, acceleration and time

velocity as a function of the initial velocity, acceleration and time

for the final velocity and the time it took to reach the ground. Notice that because your position is decreasing (from your initial height to zero), your final velocity will be negative.

What happens when you hit the ground seems more complicated because it happens so quickly, but the same equations can be used here as well. Just as you first touch the ground, and before the ground begins to reduce your velocity, your velocity is equal to the final velocity you computed for the end of your fall. Thus your final velocity from the previous computation becomes the initial velocity for this computation. In a very short period of time, the ground reduces your velocity to zero and at the same time, your body compresses the ground slightly. Therefore in this computation, your final velocity is zero and your change in position is equal to the amount the ground was compressed. It is important to note that the acceleration is no longer equal to -g; you will have to solve for the acceleration in this part of the problem. Since your final velocity is zero below ground level, your final position will be negative, and since your velocity went from negative to zero, your acceleration will be positive (even though you might think of it as deceleration, we will not use that term).

From an algebraic point of view nothing has changed:

in analyzing your fall, you were given the initial and final positions, the initial velocity and the acceleration;

in analyzing your stop, you are given the initial position (y = 0), the time it takes you to stop (which will be given to you by the applet) and the initial and final velocities.

So in each case you have a system of two equations in six variables (initial conditions count as variables for algebraic purposes):
initial position, final position, initial velocity, final velocity, acceleration and time.
And in each case, you are given the values of four of the variables. This means that you are left with a system of two equations with two unknowns, which of course has a unique solution.

This applet will give you the initial height and ask you for the time of fall and final velocity. Then it will give you the time it takes you to stop and ask you for the braking distance, or change in position when you stop, and the acceleration as you stop. Note that the braking distance is positive since in this part of the problem

Δy = 0 - yfinal
and yfinal is negative. All answers are entered in scientific notation (as x * 10y, where x >= 1 and x < 10), and the correct units must be supplied (m, s, m/s or m/s^2). Note that the applets use the symbol "^" to denote exponentiation.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at

The next section extends our work in this section to 2 dimensions.

©2003, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

Please send comments or suggestions to the author.