91Ó°ÊÓ

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.

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.

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.