NUMB3RS - Season 2 - "Assassin" - Experimenting with Paper Airplanes

The study of how objects move through the air, known as aerodynamics, relies on a balance of four primary forces:

• Thrust: The forward force pushing the plane (e.g., your throw).
• Drag: The air resistance working against the forward motion.
• Lift: The upward aerodynamic force generated by air flowing under and over the wings.
• Gravity: The downward pull of Earth's mass.

For a paper airplane, stability depends heavily on its center of gravity (weight distribution). If the nose is too light, the plane will pitch up and stall; if too heavy, it will nose-dive.

To conduct a rigorous experiment, you must isolate and control these variables. For instance, you can alter your launch angle (low angles < 45^{\circ} vs. high angles between $45^{\circ}$ and $90^{\circ}$) or shift the center of gravity by adding paperclips to the nose.

The Reynolds Number ($R$) is modeled in air by the formula:

$R = K \cdot V \cdot L$

Where:

• $K = 9346$ (a physical constant for air)
• $V$ is the velocity of the aircraft in miles per hour (mph)
• $L$ is the wing width from front to back at the middle of the wing (in feet)

Let's apply this formula to a light recreational plane: the VP-2. This aircraft cruises at a velocity $V = 87$ mph with a wing width $L = 4.815$ feet.

What is the Reynolds Number ($R$) of the VP-2?

We can also rearrange our formula algebraically to solve for other flight parameters.

Suppose a Cessna 150 has a Reynolds Number $R = 4,833,562$ while cruising at a velocity $V = 108$ mph.

Using the formula $R = K \cdot V \cdot L$ (with $K = 9346$), find the wing width $L$ in feet.

A Cherokee Cruiser has a wing width $L = 4.857$ feet and a calculated Reynolds Number $R = 5,615,547$ under standard cruising conditions.

Calculate its cruise velocity $V$ in miles per hour. Round your answer to the nearest whole number.

Let's examine how the variable parameters behave together.

Consider the relationship between the product $V \cdot L$ (velocity times wing length) and the Reynolds number $R$.

Since $R = 9346 \cdot (V \cdot L)$, what type of variation/proportion exists between $V \cdot L$ and $R$?

If you were to plot the Reynolds Number ($R$) on the vertical axis (y-axis) against the product of velocity and wing length ($V \cdot L$) on the horizontal axis (x-axis), what would the resulting graph look like?

Because the equation simplifies to $y = 9346x$, the graph is a straight line passing through the origin with a very steep positive slope of $9346$.

You can use the pre-loaded Desmos graphing utility above to visualize this relationship and see how quickly the Reynolds Number scales as the product of wing dimension and velocity increases!