Question

The velocity of a skydiver (in \(\mathrm{m} / \mathrm{s}\) ) \(t\) seconds after jumping from the plane is \(v(t)=600\left(1-e^{-k t / 60}\right) / k\), where \(k > 0\) is a constant. The terminal velocity of the skydiver is the value that \(v(t)\) approaches as \(t\) becomes large. Graph \(v\) with \(k=11\) and estimate the terminal velocity.

Short answer

Answer: The terminal velocity of the skydiver is approximately 54.54 m/s.

Step-by-step solution

Identify the given information

The given function is \(v(t)= \frac{600(1-e^{-kt/60})}{k}\), where \(k = 11\).

Plug in the given value of k

Replace \(k\) with \(11\) in the given function: \(v(t)= \frac{600(1-e^{-11t/60})}{11}\)

Find the limit as t approaches infinity

To find the terminal velocity, we need to find the limit of \(v(t)\) as \(t\) approaches infinity: \(\lim_{t \to \infty} \frac{600(1-e^{-11t/60})}{11}\) As \(t\) approaches infinity, the exponent in the \(e^{-11t/60}\) term becomes very large and negative, so the value of \(e^{-11t/60}\) approaches zero. Then: \(\lim_{t \to \infty} \frac{600(1-e^{-11t/60})}{11} = \frac{600(1 - 0)}{11}\)

Evaluate the limit

Now, we evaluate the limit: \(\frac{600(1 - 0)}{11} = \frac{600}{11} \approx 54.54 \, \mathrm{m/s}\) The terminal velocity of the skydiver is approximately \(54.54 \, \mathrm{m/s}\)

Graph the function

To graph the function, we want to plot \(v(t)\) for a range of values of \(t\). Graphing software or a graphing calculator can be used to plot \(v(t) = \frac{600(1-e^{-11t/60})}{11}\) for \(0 \leq t \leq 100\) seconds.

Estimate the terminal velocity from the graph

By observing the graph of \(v(t)\), it becomes clear that the function approaches a horizontal asymptote at around \(54.54 \, \mathrm{m/s}\) as \(t\) becomes large. This confirms our earlier calculation of the terminal velocity. In conclusion, the terminal velocity of the skydiver is approximately \(54.54 \, \mathrm{m/s}\).

Key concepts

Velocity

Velocity is an important concept in calculus, especially when analyzing motion, like that of a skydiver. Velocity is defined as the rate of change of position with respect to time. It provides both the speed of an object and the direction of its movement. In the case of the skydiver, we are given a velocity function, \(v(t)\), which describes how the velocity changes over time. This velocity function is instrumental in predicting the skydiver's behavior during the fall.

• Velocity is not just speed; it is speed with direction.
• This function helps us understand how quickly the skydiver reaches terminal velocity.

Understanding velocity allows us to grasp how forces like gravity affect the motion at various points.
It explains how quickly the skydiver reaches equilibrium with air resistance.
Hence, the direction and rate of change of velocity are crucial for determining motion patterns.

Limit

The concept of a limit is central to calculus, and it helps us understand behavior as a function approaches a certain point. In the context of our skydiver, we use limits to determine the terminal velocity, which is the velocity the skydiver eventually maintains as time becomes very large. This is where the rate of acceleration due to gravity equals the rate of resistance by air.

• Limits help us find values even if the function doesn't "reach" that point.
• They are fundamental in calculating terminal conditions, such as velocity.

Mathematically, as we calculate the limit of \(v(t)\) as \(t\) approaches infinity, we discover the terminal velocity.
We observe function behavior and essentially predict how it "settles" over time.
This limit shows us that the velocity levels off, indicating balance between competing forces.

Exponential Function

Exponential functions are a staple in calculus, modeling growth or decay processes. In the velocity equation \(v(t)\), the term \(e^{-kt/60}\) represents exponential decay.
This part of the function models how the deceleration effects reduce as time progresses.

Understanding the Decay

The negative exponent in the expression indicates decay.

• Decaying exponentials like \(e^{-11t/60}\) approach zero as \(t\) approaches infinity.
• This property is crucial in showing how the skydiver's velocity stabilizes.

Here, the exponential decay allows us to compute the diminishing effect of initial forces.
It's about how quickly the skydiver entices equilibrium with terminal velocity.
Such functions enable real-world applications by predicting short and long-term states of systems.

Graphing Functions

Graphing is an essential skill for visualizing functions and understanding their behavior. By graphing \(v(t)\), we can visually approximate the terminal velocity of our skydiver.
In graphing, we plot various values of a function over a range and observe tendencies and patterns.

Significance of the Graph

Creating a graph for \(v(t) = \frac{600(1-e^{-11t/60})}{11}\) from \(0\) to \(100\) seconds allows us to see how quickly the velocity changes over time.

• Graphs visually show asymptotic behavior, where a function approaches a stable value.
• Graphing the function helps identify the terminal velocity visually as a horizontal asymptote.

By interpreting these visual cues correctly, we can confirm our calculations.
The graph essentially allows us to "see" behavior over time in motion and rate of reach towards terminal speed.
Graphs are powerful tools in calculus to show trends or constants within dynamic systems.