Problem 69 - Calculus: Early Transcendentals

Question

In Chapter 9 we will be able to show, under certain assumptions, that the velocity \( v(t) \) of a falling raindrop at time \( t \) is $$ v(t) = v^*(1 - e^{-gt/v^*}) $$ where \( g \) is the acceleration due to gravity and \( v^* \) is the \(terminal \) \( velocity \) of the raindrop. (a) Find \( \displaystyle \lim_{t \to \infty} v(t) \). (b) Graph \( v(t) \) if \( v^* = 1 m/s \) and \( g = 9.8 m/s^2 \). How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

Step-by-Step Solution

Answer

(a) \( v^* \). (b) See graph; \( t \approx 0.47 \) seconds to reach 99% of \( v^* \).

Step 1: Determine Limit for Velocity

To find \( \lim_{t \to \infty} v(t) \), we need to analyze the limit of \( v(t) = v^*(1 - e^{-gt/v^*}) \) as \( t \) approaches infinity. As \( t \to \infty \), \( e^{-gt/v^*} \to 0 \). Therefore, the expression simplifies to \( v(t) = v^*(1 - 0) = v^* \). Thus, \( \lim_{t \to \infty} v(t) = v^* \).

Step 2: Graph the Velocity Function

Using the given values \( v^* = 1 \text{ m/s} \) and \( g = 9.8 \text{ m/s}^2 \), the velocity function becomes \( v(t) = 1(1 - e^{-9.8t}) \). To graph \( v(t) \), plot \( v(t) \) against \( t \) starting from \( t = 0 \). Initially, \( v(t) \) starts at 0, and as \( t \) increases, \( v(t) \) approaches \( 1 \) m/s asymptotically.

Step 3: Calculate Time to 99% Velocity

To find when \( v(t) = 0.99v^* \), solve \( 1(1 - e^{-9.8t}) = 0.99 \). This simplifies to \( 1 - e^{-9.8t} = 0.99 \), so \( e^{-9.8t} = 0.01 \). Taking the natural logarithm of both sides, we find \( -9.8t = \ln(0.01) \), leading to \( t = -\frac{\ln(0.01)}{9.8} \). Calculating this gives \( t \approx 0.47 \) seconds.

Key Concepts

Limits of a Function

When dealing with functions like the velocity of a raindrop over time, it’s crucial to understand limits. A limit is a way to describe the behavior of a function as the input approaches a specific value, often infinity. In our problem, we want to find out what happens to the velocity function \( v(t) = v^*(1 - e^{-gt/v^*}) \) as time \( t \) increases to infinity.

This involves understanding exponential decay within the function. As \( t \) approaches infinity, \( e^{-gt/v^*} \) approaches zero. This means the expression for velocity simplifies to the raindrop's terminal velocity, \( v^* \). Therefore, we say that the limit of \( v(t) \) as \( t \to \infty \) is \( v^* \).

To visualize, imagine the raindrop accelerating until it reaches a speed where the drag force equalizes the gravitational pull. This constant speed is the terminal velocity, and here it's confirmed by \(\lim_{t \to \infty} v(t) = v^*\). Understanding limits helps predict long-term behavior of functions.

Exponential Decay

Exponential decay is a fundamental concept in mathematics which describes the process of reducing an amount by a consistent percentage rate over a period of time. In the velocity formula for the raindrop, \( e^{-gt/v^*} \) represents exponential decay.

As time passes, \( e^{-gt/v^*} \) decreases. This term reflects how the force opposing the raindrop’s fall reduces its acceleration over time.

The connection here is key: as the exponent in \( e^{-gt/v^*} \) becomes more negative with larger values of \( t \), the entire \( e^{-gt/v^*} \) approaches zero. This decay within the function means that the velocity of the raindrop gradually stops increasing and settles at the terminal velocity \( v^* \).

Exponential decay allows phenomena like terminal velocity to appear as they smoothly transition towards a constant value. This concept is especially useful when predicting behavior over long periods.

Graphing Functions

Graphing functions like the velocity of a raindrop provides a visual representation of its behavior over time. For the given function \( v(t) = v^*(1 - e^{-gt/v^*}) \), graphing helps illuminate how the velocity changes as time progresses.

In this case, using the parameters \( v^* = 1 \text{ m/s} \) and \( g = 9.8 \text{ m/s}^2 \), we plot \( v(t) \) against \( t \).

Initially, the graph of \( v(t) \) starts at zero when \( t = 0 \). As time increases, the graph shows the velocity gradually rising and approaching \( 1 \text{ m/s} \) asymptotically. This means that the graph gets very close to the \( 1 \text{ m/s} \) line but never quite reaches it.

The crucial point on this graph is the time it takes to reach 99% of terminal velocity. Calculated as approximately 0.47 seconds, this highlights how quickly the raindrop reaches near-constant speed. Crafting such graphs provides deeper insights into dynamic processes and serves as a vital tool for understanding how functions evolve over time.