The dependence of air resistance on velocity

The force of air resistance clearly depends on the velocity of an object moving through the air: the larger the speed, the larger the drag force. But what is the exact form of this relationship?

Some textbooks suggest that under some circumstances, air resistance depends on the square of the velocity:

However, other sources suggest that at low speeds, the air resistance grows linearly with velocity:

But -- is it possible that this relationship has a form with a non-integer exponent?

Your job today is to figure out which of these formulae more accurately fits the data from a simple experiment.

The experiment

Each team will make measurements of the same event: I'll drop one set of filters, and the entire class will share all the resulting measurements.

• Create a set of objects with the same size and shape, but different mass, by placing paper clips. We'll use 0, 1, 2, 4, 6, and 8 clips placed into a single filter.
• Give the objects what we HOPE will be terminal velocity by dropping them from the third floor of the atrium. After a short acceleration, each one will (we hope) reach a constant speed for the majority of its fall.
• Have a team member stand on the staircase in the middle of the atrium and measure the time it takes for each stack to fall from the top of the first-floor railing to the bottom of the atrium.
• Make two trials for each stack.
• Calculate the speed of each stack during this final portion of its flight; the distance from atrium floor to first-floor railing is 5.8 meters.

You'll need to know the masses of the objects involved:

• the mass of one filter is 1.60 grams
• the mass of one paper clip is 0.38 grams

The model

When an object has reached terminal velocity, the downward pull of gravity exactly balances the upward push of air resistance:

That means that you can calculate the force of air resistance easily, if you know the mass of the falling filters.

Now, suppose that the relationship between between the force of air resistance and velocity looks like this:

A good way to see if measurements match a theory is to make a graph on which the theory predicts there should be a straight line:

But how can we make a graph on which the data will fall in a straight line when we don't even know the value of the exponent p?

Use LOGARITHMS!

Log-log plots and power laws

When physicists think that one quantity depends on some other quantity raised to a power, they often turn to log-log graphs. Starting with a formula in which the force is proportional to some power p of the velocity, take the logarithm of both sides, and then make a graph based on that new equation.

1. Make a graph based on the final form of the equation above; you will need to make a table showing the logarithms of your measurements.
2. Fit a straight line to the data in this log-log graph. What is the equation of the line? What is the value of the power p based on this method?
3. Which of our simple models -- air resistance going like velocity or velocity-squared -- does this method support?