Problem 66 - Single Variable Calculus

Question

The velocity of a body of mass \(m\) falling from rest through a viscous medium is given by $$ v(t)=\sqrt{\frac{m g}{k}} \tanh \left(\sqrt{\frac{g k}{m}} t\right) $$ where \(g\) is the acceleration of gravity and \(k\) is a positive constant that depends on the viscosity of the medium. a. Find \(\lim _{t \rightarrow \infty} v(t)\). b. Plot the graph of \(v\) taking \(m=2, g=32\), and \(k=8\). Note: This limiting velocity of the body is called the terminal velocity

Step-by-Step Solution

Answer

Short Answer: a. The terminal velocity, or the limit of velocity \(v(t)\) as time goes to infinity, can be found as follows: \(v_{terminal} = \lim_{t \rightarrow \infty} v(t) = \sqrt{\frac{mg}{k}}\) b. After substituting the given values, our velocity function becomes: \(v(t) = 4 \tanh (8t)\) To plot the graph, use a graphing software or an online tool, and you should observe that as time goes to infinity, the velocity of the body approaches the terminal velocity, which in this case is 4 units.

Step 1: Write down the given function

We have the velocity function given as: \(v(t) = \sqrt{\frac{mg}{k}} \tanh \left(\sqrt{\frac{gk}{m}} t\right)\)

Step 2: Analyze the behavior of the hyperbolic tangent function

As the argument of the hyperbolic tangent goes to infinity, the hyperbolic tangent itself approaches 1. Mathematically, we have: \(\lim_{x \rightarrow \infty} \tanh(x) = 1\)

Step 3: Apply the limit to v(t)

Now, we need to apply the limit to the function \(v(t)\): \(\lim_{t \rightarrow \infty} v(t) = \lim_{t \rightarrow \infty} \sqrt{\frac{mg}{k}} \tanh \left(\sqrt{\frac{gk}{m}} t\right)\) Since we know that \(\lim_{x \rightarrow \infty} \tanh(x) = 1\), we substitute 1 for the hyperbolic tangent in the limit: \(\lim_{t \rightarrow \infty} v(t) = \sqrt{\frac{mg}{k}}(1)\) Thus, the terminal velocity is given by: \(v_{terminal} = \sqrt{\frac{mg}{k}}\) #b. Plotting the graph of v(t) with given values#

Step 4: Substitute the given values

We are given the values of \(m=2, g=32\), and \(k=8\). Substituting these into the equation of \(v(t)\), we get: \(v(t) = \sqrt{\frac{2(32)}{8}} \tanh \left(\sqrt{\frac{32(8)}{2}} t\right)\) Simplifying the equation, we obtain: \(v(t) = 4 \tanh (8t)\)

Step 5: Create the plot

Using the simplified equation for \(v(t)\), we will now create a plot of the function with respect to time (\(t\)). You can use any graphing software or online tool to plot the graph. The graph will show the velocity of the object increasing with time until it reaches its terminal velocity, which we found in part a. You should observe that as time goes to infinity, the velocity of the body approaches the terminal velocity of 4 units, which is consistent with our findings in part a.

Key Concepts

Viscous Medium

A viscous medium is a type of environment that slows down the motion of objects moving through it, due to its internal friction. Think of honey or thick oil as common examples where viscosity plays a crucial role.
When a body moves through such a medium, resistance is encountered. This force opposing the motion is a type of friction called viscous drag. It acts in the opposite direction to the motion.

• Viscosity is like the thickness of a fluid. The higher it is, the slower objects will move through it.
• In our problem, the constant \(k\) is directly related to the viscosity of the medium.

Understanding a viscous medium is important because it helps in calculating the terminal velocity. Terminal velocity is the maximum speed an object will reach when the force of gravity is balanced by the drag force from the viscous medium.

Hyperbolic Tangent Function

The hyperbolic tangent function, written as \(\tanh(x)\), is a mathematical function that behaves like the traditional tangent but is used for hyperbolic lines.
It has properties similar to exponential functions and can be thought of as a smoother version of sine and cosine functions.

• The formula used in our problem is \(\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}\).
• Key property: As \(x\) increases towards infinity, \(\tanh(x)\) approaches 1.

In the context of the velocity function, \(\tanh(x)\) helps in modeling how rapidly motion approaches terminal velocity. The function gradually transitions, allowing for smooth and predictable behavior over time.

Limit of a Function

The limit of a function is a fundamental concept in calculus. It describes the value that a function approaches as the input (or independent variable) gets infinitely close to some number.
In the velocity problem, we are interested in the limit of \(v(t)\) as \(t\) approaches infinity, meaning, as time goes on indefinitely.

• We calculated that \(\lim_{t \to \infty} v(t) = \sqrt{\frac{mg}{k}}\).
• This calculation shows that no matter how much more time passes, the velocity will level off at this value. This is known as terminal velocity.

The limit helps us understand long-term behavior without having to simulate every moment in time, providing a snapshot of infinite progression in an instant.

Plotting Graphs

Plotting graphs allows us to visually interpret functions and understand their behavior over a range of values. This can be incredibly informative, providing clarity that might not be as apparent by just looking at the equations.
In our specific exercise, plotting \(v(t) = 4 \tanh(8t)\) helps illustrate how velocity changes over time.

• Tools like graphing calculators, software (e.g., Desmos, GeoGebra), or coding libraries (e.g., Matplotlib in Python) can be used for plotting.
• The graph shows a curve where velocity starts at zero, then quickly increases before leveling off at the terminal velocity we calculated, which is 4 units.

Visualizing this curve helps confirm that the velocity indeed approaches the value calculated using the limit, providing a complete understanding of the motion through the viscous medium.