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Experimental determination of the drag coefficient: When retardation due to air resistance is proportional to downward velocity \(V\), in feet per second, falling objects obey the equation of change $$ \frac{d V}{d t}=32-r V $$ where \(r\) is known as the drag coefficient. One way to measure the drag coefficient is to measure and record terminal velocity. a. We know that an average-size man has a terminal velocity of 176 feet per second. Use this to show that the value of the drag coefficient is \(r=0.1818\) per second. (Hint: To say that the terminal velocity is 176 feet per second means that when the velocity \(V\) is 176 , velocity will not change. That is, \(\frac{d V}{d t}=0\). Put these bits of information into the equation of change and solve for \(r\).) b. An ordinary coffee filter has a terminal velocity of about 4 feet per second. What is the drag coefficient for a coffee filter?

Step by step solution

01

Understanding Terminal Velocity

Terminal velocity is reached when the velocity is constant, meaning the acceleration is zero. For this problem, when velocity reaches 176 feet per second for an average-sized man, we have \(\frac{d V}{d t} = 0\).

02

Substituting Terminal Velocity into the Equation

Given that terminal velocity \(V = 176\) and \(\frac{d V}{d t} = 0\), substitute these values into the equation \(\frac{d V}{d t} = 32 - r V\) yielding: \[0 = 32 - r \times 176.\]

03

Solving for the Drag Coefficient r for Part (a)

Rearrange the equation \(0 = 32 - 176r\) to solve for \(r\): \[ 176r = 32\] \[ r = \frac{32}{176} = 0.1818 \text{ per second.} \]

04

Calculating the Drag Coefficient for a Coffee Filter (Part b)

Similarly, for a coffee filter with a terminal velocity of 4 feet per second, substitute \(V = 4\) and \(\frac{d V}{d t} = 0\) into the equation: \[ 0 = 32 - 4r. \]

05

Solving for the Drag Coefficient for a Coffee Filter

Rearrange the equation \(0 = 32 - 4r\) to solve for \(r\): \[ 4r = 32 \] \[ r = \frac{32}{4} = 8 \text{ per second.} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Drag Coefficient

The drag coefficient (r) is a crucial factor in determining how various objects move through a fluid, like air. It is a measure of the amount of air resistance an object encounters as it falls. The drag coefficient is not a fixed number - it varies based on several factors like

• the shape of the object,
• the surface roughness,
• and the fluid's properties (in this case, air).

When an object reaches terminal velocity, its downward motion is constant due to the balance of gravitational force and air resistance, symbolizing stability in velocity where \( \frac{d V}{d t} = 0 \). This constant velocity can tell us about the drag coefficient using the differential equation given, allowing us to see the resistance faced by diverse objects such as a human or a coffee filter. For instance, using experiments, we can find this coefficient by comparing different objects and their terminal velocities.

Differential Equation

Differential equations are mathematical expressions that show the relationship between a function and its derivatives, helping us understand how a quantity changes over time. In our case, the equation \( \frac{d V}{d t} = 32 - r V \) demonstrates how the velocity (V) of objects changes with time due to air resistance.
It captures two important aspects: one is the constant gravitational pull (represented by 32, as acceleration due to gravity), and the other is the resisting drag effect \( - rV \). This opposing force decreases the velocity incrementally until a balance is reached at terminal velocity, as observed when \( \frac{d V}{d t} = 0 \).
By substituting known values of terminal velocity into this equation, we can solve for the drag coefficient r, visually showing how the drag force balances the gravitational pull at terminal velocity.

Air Resistance

Air resistance, also known as drag force, is a force acting opposite to the relative motion of any object moving through the air. It opposes the downward pull of gravity when an object is falling, causing it to eventually reach a stable terminal velocity instead of accelerating indefinitely. Air resistance depends on factors like:

• velocity of the object,
• surface area,
• and the air's density.

As an object speeds up like when falling, the drag force becomes larger until it equals gravitational force. This balance of forces at terminal velocity means that all acceleration halts and the object continues to fall at a stable speed.
The concept of air resistance is fundamental in understanding how differently shaped objects, such as people or simple objects like coffee filters, have varying terminal velocities, illustrating how nature acts to "slow down" objects in motion through air.