Problem 46 - Essential Calculus Early Transcendentals

Question

If an object with mass \(m\) is dropped from rest, one model for its speed \(v\) after \(t\) seconds, taking air resistance into account, is $$v=\frac{m g}{c}\left(1-e^{-c t / m}\right)$$ where \(g\) is the acceleration due to gravity and \(c\) is a positive constant. (a) Calculate \(\lim _{t \rightarrow \infty} v .\) What is the meaning of this limit? (b) For fixed \(t,\) use l'Hospital's Rule to calculate lim \(_{c \rightarrow 0^{+}} v\) What can you conclude about the velocity of a falling object in a vacuum?

Step-by-Step Solution

Answer

(a) \(\frac{mg}{c}\) - terminal velocity. (b) \(gt\) - velocity in vacuum.

Step 1: Understand the Problem

We are given a formula for velocity taking air resistance into account: \(v = \frac{m g}{c} \left(1 - e^{-ct/m}\right)\). We need to compute two specific limits and interpret them. One is as time \(t\) approaches infinity, and the other as the air resistance constant \(c\) approaches zero.

Step 2: Calculate Limit as t Approaches Infinity

To find \(\lim_{t \to \infty} v\), substitute \(v = \frac{mg}{c}(1 - e^{-ct/m})\) into the limit. As \(t\) approaches infinity, \(e^{-ct/m}\) approaches zero, resulting in \(v = \frac{mg}{c}(1 - 0) = \frac{mg}{c}\). This shows that the object reaches a terminal velocity of \(\frac{mg}{c}\) due to air resistance over time.

Step 3: Interpret the Meaning of Limit (t → ∞)

The limit \(\lim_{t \to \infty} v = \frac{mg}{c}\) represents the terminal velocity of the object. Terminal velocity is the constant speed that the object reaches when the force of gravity is balanced by air resistance.

Step 4: Calculate Limit as c Approaches Zero Using L'Hospital's Rule

We apply L'Hospital's Rule to \(\lim_{c \to 0^+} \frac{mg}{c}(1 - e^{-ct/m})\) since it is in an indeterminate form \(\frac{0}{0}\). Differentiating the numerator and denominator gives \(\lim_{c \to 0^+} \frac{d}{dc}[mg(1 - e^{-ct/m})] / \frac{d}{dc}[c]\). After differentiation, the limit simplifies to \(g t\).

Step 5: Interpret the Meaning of Limit (c → 0)

The limit \(\lim_{c \to 0^+} v = g t\) indicates that in a vacuum (i.e., when air resistance is negligible), the velocity of the object is linearly proportional to time, which is consistent with the acceleration due to gravity.

Key Concepts

Air Resistance

Air resistance, also known as drag, is a force that opposes the motion of an object through a fluid, such as air. It plays a crucial role in determining the speed at which an object falls. When an object moves through air, it must push air molecules out of the way, and this interaction creates resistance. The greater the speed of the object, the more air it must displace, leading to increased resistance.

There are several factors that affect air resistance:

• Velocity of the Object: As the velocity increases, so does the air resistance, which acts in the opposite direction to slow down the object.
• Cross-Sectional Area: A larger area means more air molecules are displaced, increasing the resistance encountered.
• Shape of the Object: Streamlined shapes experience less air resistance compared to more blunt shapes.
• Surface Roughness: A smoother surface reduces the friction with air molecules, lowering resistance.

Air resistance becomes significant over time and eventually balances the force of gravity, preventing the object from accelerating further. This balance leads to a state called terminal velocity.

Terminal Velocity

Terminal velocity is the steady speed attained by an object freely falling through a fluid under the uniform force of gravity. This occurs when the downward gravitational force equals the upward air resistance force, resulting in zero net acceleration. Consequently, the object continues to fall at a consistent speed.

In the given exercise, when we find the limit of velocity as time approaches infinity, we get the formula for terminal velocity: \( v = \frac{mg}{c} \).
This tells us that terminal velocity depends on several factors:

• Mass of the Object (\(m\)): Heavier objects generally reach higher terminal velocities because the gravitational force is stronger.
• Air Resistance Constant (\(c\)): A higher value means more resistance, leading to a lower terminal velocity.
• Gravity (\(g\)): The greater the gravitational pull, the higher the terminal velocity.

Understanding terminal velocity helps in applications like designing parachutes, which are meant to increase air resistance and lower the terminal speed for a safe landing.

l'Hospital's Rule

L'Hospital's Rule is a powerful tool in calculus used to evaluate limits that result in indeterminate forms. Indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) arise when evaluating certain limits, and l'Hospital's Rule provides a way to resolve these forms by differentiating the numerator and denominator separately.

In our exercise, we used l'Hospital’s Rule to analyze the limit as the air resistance constant \(c\) approaches zero in the formula. The expression became indeterminate in the form of \(\frac{0}{0}\), making it an ideal candidate for this rule.

By applying l'Hospital's Rule, we found that \[ \lim_{c \to 0^+} \frac{mg}{c}(1 - e^{-ct/m}) = gt \]. This shows that as air resistance becomes negligible (similar to a vacuum), the velocity of a falling object in terms of time is directly related to gravity and time, indicating a linear relationship: velocity increases steadily as time increases.