Problem 8 - Active Calculus

Question

A bungee jumper dives from a tower at time \(t=0\). Her height \(s\) in feet at time \(t\) in seconds is given by \(s(t)=100 \cos (0.75 t) \cdot e^{-0.2 t}+100\). a. Write an expression for the average velocity of the bungee jumper on the interval \([1,1+\) \(h]\) b. Use computing technology to estimate the value of the limit as \(h \rightarrow 0\) of the quantity you found in (a). c. What is the meaning of the value of the limit in (b)? What are its units?

Step-by-Step Solution

Answer

The expression for average velocity is \(\frac{s(1 + h) - s(1)}{h}\). The estimated limit as \(h \rightarrow 0\) is the instantaneous velocity at \(t = 1\) second, with units of \(\text{ft/s}\).

Step 1 - Understand the given function

The height function of the bungee jumper is given by: \[ s(t) = 100 \cos (0.75 t) \cdot e^{-0.2 t} + 100 \] It represents her height in feet at time t in seconds.

Step 2 - Write the expression for average velocity

The average velocity \(v\text{avg}\) on the interval \[1, 1 + h\] is given by: \[ v\text{avg} = \frac{s(1 + h) - s(1)}{h} \] To find this, we'll need to substitute into our height function.

Step 3 - Calculate \(s(1)\)

First, we'll calculate the height at \(t=1\): \[ s(1) = 100 \cos (0.75 \cdot 1) \cdot e^{-0.2 \cdot 1} + 100 \] Compute this using a calculator to get a numerical value.

Step 4 - Calculate \(s(1 + h)\)

Next, calculate the height at \(t=1+h\): \[ s(1+h) = 100 \cos (0.75 (1 + h)) \cdot e^{-0.2 (1 + h)} + 100 \] This is a bit more complex; simplify the expression as much as possible.

Step 5 - Form the average velocity expression

Substitute \(s(1)\) and \(s(1+h)\) into the average velocity formula: \[ \frac{100 \cos (0.75 (1 + h)) \cdot e^{-0.2 (1 + h)} + 100 - (100 \cos (0.75 \cdot 1) \cdot e^{-0.2 \cdot 1} + 100)}{h} \]

Step 6 - Simplify the average velocity expression

The expression simplifies to: \[ v\text{avg} = \frac{100 ( \cos (0.75 (1 + h)) \cdot e^{-0.2 (1 + h)} - \cos (0.75) \cdot e^{-0.2} )}{h} \]

Step 7 - Estimate the limit as \(h \rightarrow 0\)

Using computing technology (a calculator or computer algebra system), estimate the limit of the expression as \(h \rightarrow 0\): \[ \lim_{{h \rightarrow 0}} \frac{s(1 + h) - s(1)}{h} \]

Step 8 - Meaning and units of the limit

The limit represents the instantaneous velocity of the bungee jumper at \(t = 1\) second. The units of the instantaneous velocity are feet per second \(\text{ft/s}\).

Key Concepts

Instantaneous Velocity

In this exercise, we are tasked with finding the average velocity of a bungee jumper and then determining its instantaneous velocity.
Instantaneous velocity is essentially the speed of an object at a single moment in time.
Mathematically, it is the limit of the average velocity as the time interval approaches zero.
The formula we use for average velocity on the interval \( [1, 1 + h] \) is: \[ v\text{{avg}} = \frac{{s(1 + h) - s(1)}}{h} \]
To find the instantaneous velocity, we calculate the limit of this expression as \( h \rightarrow 0 \).
This value tells us how fast the bungee jumper is moving at exactly \( t = 1 \) second. The process involves a limit calculation, which we will elaborate on below. It’s important to understand that instantaneous velocity gives a precise indication of the bungee jumper’s speed at a specific instant, unlike average velocity which is over a range.

Bungee Jumper Height Function

The height function provided in the exercise is critical for calculating both average and instantaneous velocities.
The height function is given as: \[ s(t) = 100 \cos(0.75 \cdot t) \cdot e^{-0.2 \cdot t} + 100 \]
This equation takes time \( t \) and calculates the height \( s \) in feet.
Elements of this function include:

• The cosine function \( \cos(0.75 \cdot t) \) which models oscillation due to bungee jumping motions.
• The exponential decay \( e^{-0.2 \cdot t} \) which represents the damping effect over time, reduction of bounce height.
• The constant 100 added at the end, representing the initial height from which the jumper starts.

Understanding these elements helps see how the height changes over time, giving insights into the dynamics of a bungee jump.

Limit Calculation

To find the instantaneous velocity, we need to compute the limit of the average velocity expression as \( h \rightarrow 0 \).
This is a fundamental concept in calculus known as taking the derivative. Here’s how we approach it:

• First, we calculate \( s(1) \) by substituting \( t = 1 \) in the height function: \[ s(1) = 100 \cos (0.75 \cdot 1) \cdot e^{-0.2 \cdot 1} + 100 \]
  Plugging this into a calculator provides a numerical result.
• Next, calculate \( s(1 + h) \), which involves substituting \( t = 1 + h \) in the height function. Because it's more complicated, simplifying numerically via a calculator or a CAS tool helps.
• Substitute these values into the average velocity formula: \[ \frac{100 \cos (0.75 (1 + h)) \cdot e^{-0.2 (1 + h)} + 100 - (100 \cos (0.75) \cdot e^{-0.2} + 100)}{h} \]
• Simplify to: \[ v\text{avg} = \frac{100 \left ( \cos (0.75 (1 + h)) \cdot e^{-0.2 (1 + h)} - \cos (0.75) \cdot e^{-0.2} \right )}{h} \]
• Estimate the limit using technology: \[ \lim_{{h \rightarrow 0}} \frac{s(1+h) - s(1)}{h} \] This limit is the instantaneous velocity at \( t = 1 \), commonly written in derivatives as \( s'(1) \).
• The result is expressed in feet per second \((\text{{ft/s}})\).

This process, though technical, enables us to understand instantaneous changes in the height function, thus giving insights into velocity at a precise moment.