Problem 72 - A Graphical Approach to Precalculus with Limits

Question

MODELING Speed of a Skydiver If air resistance is not ignored during free fall, then the speed (in feet per second) of a skydiver after \(t\) seconds is given by $$ f(t)=176\left(1-e^{-a 2 t}\right) $$ Calculate \(\lim _{t \rightarrow x^{\infty}} f(t)\) and give an interpretation of its value. (IMAGES CANNOT COPY).

Step-by-Step Solution

Answer

The limit is 176; it represents the skydiver's terminal velocity.

Step 1: Understanding the Limit

The problem asks for the limit of the function \( f(t) = 176(1 - e^{-a2t}) \) as \( t \) approaches infinity. The goal is to find the behavior of the function when \( t \) becomes very large.

Step 2: Analyzing the Exponential Term

As \( t \to \infty \), the term \( e^{-a2t} \) will tend to zero because \( e^{-x} \to 0 \) as \( x \to \infty \). This is a key property of exponential decay.

Step 3: Substitute and Simplify

By substituting as \( t \to \infty \), the expression becomes: \[ f(t) = 176 \times (1 - 0) = 176. \] Thus, the limit is 176.

Step 4: Interpretation of the Limit

The limit of 176 represents the terminal velocity of the skydiver. This is the maximum speed the skydiver can reach when air resistance balances out gravitational acceleration.

Key Concepts

Exponential Decay

In many natural processes, exponential decay plays a crucial role in governing how quickly things change over time. It appears when something decreases at a rate proportional to its current value. A primary example of this is seen in the decay term of the function, where we use the expression \(e^{-a2t}\).
This expression indicates that as time \(t\) increases, the term \(e^{-a2t}\) becomes very small, closely approaching zero. This happens because the exponent is negative and increasingly large, driving the entire term toward zero.

• Exponential decay ensures that quantities decrease rapidly at first, but the rate of decrease slows over time.
• The remaining term in the function, \(176(1 - e^{-a2t})\), allows us to determine the speed of the skydiver.
• The "\(-a2t\)" signifies how powerful the decay effect is, which depends on the specific constant values like air resistance and other dynamic variables.

The understanding of exponential decay is essential to grasp how processes diminish over time, including a skydiver's velocity as air resistance takes effect.

Terminal Velocity

Terminal velocity refers to the constant speed that a falling object eventually reaches when the force of gravity is balanced by the resistive forces, like air resistance. In the scenario of skydivers falling through the air, terminal velocity is reached when air resistance negates further acceleration due to gravity.
This is evident in the limit of the skydiver's speed function, where as time progresses towards infinity, the function's value stabilizes. The formula \( f(t) = 176 \times (1 - e^{-a2t}) \) simplifies to \(176\) as the exponential term vanishes. Here's what this means:

• The speed settles at 176 feet per second, which is the terminal velocity.
• The balance occurs because the accelerating force of gravity is opposed by the increasing force of air resistance.
• This maximum speed is unique to each scenario, factoring in mass and surface area.

Understanding terminal velocity is important as it highlights how continuous acceleration isn't realistic due to resistance in real-life scenarios.

Free Fall Dynamics

Free fall dynamics describes the motion of objects when gravity is the only force acting on them. However, in real-world conditions, air resistance significantly affects this motion.
In the case of a skydiver, the free fall is modified by factors like air friction, leading to the actual movement modeled by functions such as \( f(t) = 176(1 - e^{-a2t}) \). Let's explore these dynamics further:

• Initially, the skydiver accelerates rapidly due to gravity when air resistance is minimal.
• As they gain speed, air resistance increases, reducing the net force and slowing acceleration.
• Eventually, terminal velocity is achieved, and speed levels off.

The mathematical model captures these aspects by incorporating exponential decay into the speed calculation, allowing accurate predictions of speed over time. This complex interplay between forces defines the skydiver's motion and emphasizes the impact of external resistive forces on falling objects.