Problem 91 - Elementary and Intermediate Algebra

Question

Before the parachute opens, a skydiver's velocity in meters per second is modeled by the function \(f(t)=50\left(1-e^{-0.2 t}\right)\) where \(f(t)\) is the velocity at time \(t .\) Find the velocity after 20 seconds of free fall.

Step-by-Step Solution

Answer

The velocity after 20 seconds is approximately 49.084 m/s.

Step 1: Identify the function and given variables

The function given is \(f(t)=50(1-e^{-0.2t})\), where \(f(t)\) represents the velocity at time \(t\), and we are required to find the velocity when \(t = 20\) seconds.

Step 2: Substitute the given time into the function

Substitute \(t = 20\) seconds into the function to find the velocity: \(f(20) = 50(1-e^{-0.2 \times 20})\).

Step 3: Calculate the exponent

Calculate the exponent of the exponential function: \(-0.2 \times 20 = -4\). Thus, the expression becomes \(e^{-4}\).

Step 4: Evaluate the exponential function

Use a calculator to evaluate \(e^{-4}\), which approximates to \(0.01832\).

Step 5: Subtract from 1

Subtract the result of the exponential function from 1: \(1 - 0.01832 = 0.98168\).

Step 6: Multiply by 50

Multiply the result by 50 to find the velocity: \(50 \times 0.98168\).

Step 7: Calculate the final velocity

Perform the multiplication: \(50 \times 0.98168 = 49.084\).

Step 8: Write the final answer

The velocity of the skydiver after 20 seconds of free fall is approximately 49.084 m/s.

Key Concepts

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the given exercise, the expression is in the form of an exponential decay function, represented by \( e^{-0.2t} \). This means as time \( t \) increases, the value of \( e^{-0.2t} \) decreases. This behavior is typical of processes like cooling, radioactive decay, and, as in this case, the slowing acceleration of the skydiver's velocity due to air resistance before reaching a stable terminal velocity.

An exponential function can be simplified using properties of exponents. For example, \( e^{0} = 1 \) and as \( t \rightarrow \infty \), \( e^{-0.2t} \rightarrow 0 \). This highlights how the velocity of the skydiver approaches a limit.

• The rapidly diminishing factor \( e^{-0.2t} \) represents how quickly the effect of increasing \( t \) is offset.
• The factor 50 outside the bracket scales the speed in meters per second.

Substitution in Functions

Substitution is a fundamental concept in mathematics where you replace a variable with a given value to calculate a result. In this exercise, we substitute \( t = 20 \) into the velocity function \( f(t) = 50(1-e^{-0.2t}) \). This process allows us to determine the specific velocity at a certain time during the free fall.

The steps for substitution include:

• Identifying the correct value to substitute into the function, which is \( t = 20 \).
• Replacing the \( t \) with 20 in the formula, turning it into \( f(20) = 50(1-e^{-0.2 \times 20}) \).
• Simplifying the equation by calculating the exponent and determining the exponential value, which then modifies the original expression.

This approach is vital for solving function-related problems as it helps pinpoint specific outputs based on input values, showing how changes in variables affect the overall result.

Free Fall Physics

Free fall occurs when an object is subjected only to gravity's force, meaning no other forces, such as air resistance, are significant. Initially, a skydiver accelerates due to gravity at approximately \(9.81\, \text{m/s}^2\). However, as they garner speed, air resistance increases, causing the acceleration to decrease until they reach terminal velocity, where forces of gravity and air resistance balance out.

In the exercise, the model \( f(t) = 50(1-e^{-0.2t}) \) illustrates this scenario. The exponential function plays a crucial role in demonstrating how velocity approaches a stable limit over time due to air resistance. Below are some key ideas to grasp:

• When starting free fall, initial acceleration is high, but quickly decreases as speed increases.
• The constant 50 in the function represents the limit or terminal velocity, implying that as \( t \rightarrow \infty \), velocity nears this value.
• This conceptually mirrors real-world skydiving, where a jumper accelerates until air resistance equals gravitational force, resulting in terminal velocity.

This exercise sheds light on physics concepts contextualized with real-life skydiving situations, elucidating how theoretical models describe empirical observations.