If we wanted to measure terminal velocity, we could see how far something falls, measure how long that takes, and divide to get the terminal velocity.

On the other hand, that method would fail if the object were still accelerating up to its terminal velocity when we took the measurement. To measure terminal velocity, we want to drop something, let it fall until it reaches its terminal velocity (or gets very close to it), and only then begin measuring distance and time for the rest of the fall.

To help with this, in this problem we'll address a different question:

"How far does an object have to fall through the air before it reaches its terminal velocity?"

We know that when we let something go, gravity pulls it to the ground. The speed of the object gets faster and faster as it falls. The rate that the speed increases is the object's acceleration. As we've seen, the acceleration of objects dropped near the surface of Earth (ignoring any wind drag) is \(g \approx 10 \;\mathrm{m/s^2}.\)

Assuming there is no air drag, the speed is \(|v| = gt,\)
where \(t\) is the time the object has been falling.

We've already estimated the terminal velocity to be \(|v_{\rm terminal}| \approx \sqrt{\dfrac{2 mg}{\rho A}}.\)

(We have left \(C_D\) out of this equation because we estimated that it was close to 1, and so doesn't have much effect.)

As a simple model, assume that an object that has dropped follows the equation \(|v| = gt\) until it reaches terminal velocity. This is not completely accurate, but okay for our estimation. Combine \(|v| = gt\) and \(v_{\rm terminal} = \sqrt{\dfrac{2 mg}{\rho A}}\) to estimate the time it takes a falling object to reach terminal velocity.

Next, multiply the terminal velocity by the time you just found. This gives an estimate for the distance \(d\) something needs to fall before reaching a speed close to its terminal velocity. You should get an equation for \(d\) in terms of \(m, \rho, A, C_D,\) and \(d\)

Find this equation, then use it to estimate how far a piece of notebook paper needs to fall through air before approximately reaching its terminal velocity.


Express your answer in centimeters

 Apr 16, 2024

Here's how to find the equation for the distance an object falls before reaching terminal velocity and estimate the distance for a piece of notebook paper


1. Combining Equations:


We are given two equations


Speed due to gravity: |v| = gt (where v is the object's speed, g is acceleration due to gravity, and t is time)


Terminal velocity: v_terminal = √(2mg / ρA) (where m is the object's mass, ρ is air density, A is the object's cross-sectional area, and C_D is the drag coefficient, assumed to be ≈ 1)


We want to find the time (t) it takes for the object's speed to reach terminal velocity (v_terminal).


2. Solving for Time:


Since the object accelerates constantly until reaching terminal velocity, we can set its speed (v) equal to the terminal velocity (v_terminal) in the first equation:


v_terminal = gt


Substitute the expression for terminal velocity from the second equation:


√(2mg / ρA) = gt


3. Square Both Sides (be cautious):


Square both sides to get rid of the square root (remembering that squaring introduces extraneous solutions, so we'll need to check for those later):


2mg / ρA = g^2 * t^2


4. Isolate Time:


Solve for time (t):


t = √(2mg / (ρA * g^2))


5. Estimating Distance:


Once we have the time (t), we can estimate the distance (d) the object falls by multiplying the terminal velocity (v_terminal) by the time (t):


d = v_terminal * t = √(2mg / ρA) * √(2mg / (ρA * g^2))


Simplify the equation:


d = √((2mg)^2 / (ρA * g^2 * ρA))


Cancel out common factors and remove the square root (since distance cannot be negative):


d = 2m / (ρA)


6. Estimating for Notebook Paper (as an example):


Let's estimate the distance for a piece of notebook paper (assuming no wind resistance and the chosen estimates for mass, density, and area):


Mass of a sheet of notebook paper (m): Assume m ≈ 0.005 kg (This is an estimate, the actual mass can vary)


Air density (ρ): ρ ≈ 1.2 kg/m³ (at sea level)


Area of a sheet of notebook paper (A): Assume a typical notebook paper size of 21.6 cm x 27.9 cm. Convert to meters: A ≈ 0.06 m x 0.08 m = 0.0048 m²


Plug these values into the equation:


d = 2 * 0.005 kg / (1.2 kg/m³ * 0.0048 m²)


d ≈ 0.83 meters

 Apr 17, 2024

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