Falling faster: speed is not distance

Set the rules first

Imagine an object released from rest in a uniform gravitational field, with no air resistance. We will use a rounded acceleration of ten meters per second squared. There is enough distance for the times we discuss. This is a calculation model, not a height for an experiment.

A change each second

In this model the downward speed increases by ten meters per second during each second. It starts at zero, reaches ten after one second, and twenty after two. Speed tells us how quickly position changes. Acceleration tells us how quickly velocity changes.

Do not swap the quantities

After two seconds, the speed is twenty meters per second. That does not mean the object traveled twenty meters during each of those seconds. It was moving more slowly earlier. A speed value is not a distance receipt for every part of the trip.

Add the distances

From rest, the distance fallen is one half times acceleration times time squared. With our rounded acceleration, the object falls five meters in the first second and twenty meters in two seconds total. That means the second second adds fifteen meters, not another five.

Pause and predict

Pause and extend the pattern to three seconds. Use speed equals ten times time. For distance, use five times time squared. Write the units beside both answers. Check that you have not given the distance formula a speed label or given the speed formula a distance label.

Two answers, two meanings

After three seconds, the speed is thirty meters per second and the distance traveled is forty five meters. The numbers answer different questions. The average speed over that whole interval is fifteen meters per second, half the final speed in this particular constant-acceleration, rest-start model.

An odd-number surprise

Here is a fun pattern. Successive one second intervals add five, fifteen, then twenty five meters in this model. Those are five times the odd numbers one, three, and five. Equal gains in speed do not produce equal distances in equal times.

Bring back the air

Air resistance usually grows as downward speed through still air grows. That changes acceleration, so our simple no-air table can stop being a useful prediction. Continue to the worksheet to state assumptions, compare speed with distance, and explain what the model can and cannot tell us.