Problem 67 - Algebra and Trigonometry

Question

Sky Diving A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is \(0.2 .\) It can be shown that the downward velocity of the sky diver at time \(t\) is given by $$ v(t)=80\left(1-e^{-0.2 t}\right) $$ where \(t\) is measured in seconds and \(v(t)\) is measured in feet per second \((\mathrm{ft} / \mathrm{s})\) (a) Find the initial velocity of the sky diver. (b) Find the velocity after 5 s and after 10 s. (c) Draw a graph of the velocity function \(v(t)\) . (d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.

Step-by-Step Solution

Answer

(a) Initial velocity is 0 ft/s. (b) After 5 s: 50.56 ft/s, after 10 s: 69.176 ft/s. (c) Graph shows exponential growth. (d) Terminal velocity is 80 ft/s.

Step 1: Find the Initial Velocity

To determine the initial velocity of the sky diver, substitute \( t = 0 \) into the velocity function \( v(t) = 80(1 - e^{-0.2t}) \). So, \( v(0) = 80(1 - e^{0}) = 80(1 - 1) = 0 \). The initial velocity is \( 0 \; \text{ft/s} \).

Step 2: Find the Velocity after 5 Seconds

Substitute \( t = 5 \) into the velocity function to find the velocity after 5 seconds. \[ v(5) = 80(1 - e^{-0.2 \times 5}) \] \[ v(5) = 80(1 - e^{-1}) \] Using a calculator, \( e^{-1} \approx 0.368 \). Thus, \[ v(5) = 80(1 - 0.368) = 80 \times 0.632 = 50.56 \; ft/s \].

Step 3: Find the Velocity after 10 Seconds

Now substitute \( t = 10 \) into the velocity function to find the velocity after 10 seconds. \[ v(10) = 80(1 - e^{-0.2 \times 10}) \] \[ v(10) = 80(1 - e^{-2}) \] For \( e^{-2} \approx 0.1353 \), the velocity is \[ v(10) = 80(1 - 0.1353) = 80 \times 0.8647 = 69.176 \; ft/s \].

Step 4: Graph the Velocity Function

To graph the function \( v(t) = 80(1 - e^{-0.2t}) \), plot points for a range of \( t \) values, such as 0, 1, 2, 5, 10, etc., using the evaluated values of \( v(t) \) from these steps. The graph will start at 0, rise sharply initially, and gradually approach 80 as \( t \rightarrow \infty \). The curve is an exponential approach to a horizontal asymptote at 80 ft/s.

Step 5: Determine the Terminal Velocity

The terminal velocity is found as \( t \rightarrow \infty \), the limit of \( v(t) \) is evaluated. \[ \lim_{t \to \infty} 80(1 - e^{-0.2t}) = 80(1 - 0) = 80 \; ft/s \]. From the graph and calculation, the terminal velocity is 80 ft/s.

Key Concepts

Initial Velocity

When we talk about initial velocity, we refer to the speed at which an object begins its movement. For the sky diver, the initial velocity is the velocity at the exact moment they jump off and begin their descent.
In our problem, we determine this by plugging in the time (\( t \) = 0) into the velocity function \( v(t) = 80(1 - e^{-0.2t}) \) . At \( t = 0 \), the equation simplifies to \( v(0) = 80(1 - e^0) \). Since \( e^0 = 1 \), the result is zero: \( v(0)= 80(1-1) = 0 \).

This means that the initial velocity of the sky diver is 0 ft/s. Essentially, they start at rest before gravity begins to act on them.

Terminal Velocity

As an object falls, it eventually reaches a speed where gravity and air resistance balance each other. This is known as the terminal velocity. For the sky diver, this is the maximum speed they achieve during free fall.

To find terminal velocity, we look at the velocity function as time (\( t \)) approaches infinity. Essentially, we look at the limit of \( v(t) \) as \( t \) becomes very large. \[ \lim_{t \to \infty} 80(1-e^{-0.2t}) = 80(1-0) = 80 \] The calculation shows that as time goes on, the velocity approaches 80 ft/s. This constant speed is the sky diver's terminal velocity. It indicates the point at which the forces are balanced, and the sky diver falls at a consistent speed without accelerating further.

Graph of a Function

A graph is a visual representation of how a particular function behaves. For the sky diver's velocity function, plotting the graph helps us understand how her speed changes over time.

The velocity function given is \( v(t) = 80(1 - e^{-0.2t}) \). To plot the graph, we calculate the velocity at different moments, like \( t = 0, 1, 2, 5, 10 \), and so on. Each calculated value represents a point on the graph.

• At \( t = 0 \), \( v(0) = 0 \) ft/s.
• At \( t = 5 \), \( v(5) \approx 50.56 \) ft/s.
• At \( t = 10 \), \( v(10) \approx 69.176 \) ft/s.

The plot will show a curve that starts at 0 ft/s, rises quickly at first, and then steadily approaches 80 ft/s. The curve levels off as it nears the terminal velocity, forming an asymptote. This visualization aids in comprehending how the velocity increases non-linearly as time goes on.

Exponential Function

An exponential function is characterized by a rapidly changing rate. In our problem, the function \( v(t) = 80(1 - e^{-0.2t}) \) represents the velocity as an exponential function of time.

The key feature of an exponential function like this is that it never quite reaches a limit in a finite amount of time. Instead, it approaches a horizontal asymptote. The function's exponential nature means the sky diver's velocity increases quickly at first but slows down the closer it gets to its cap, the terminal velocity.

• Exponential decay occurs due to the \( e^{-0.2t} \) part, which diminishes as \( t \) increases.
• Given \( 1-e^{-0.2t} \) means the function starts with a steep slope and flattens over time.
• This behavior is why the sky diver hits 80 ft/s gradually rather than immediately.

Understanding this helps in knowing why natural processes, like falling with air resistance, often follow exponential patterns.