Problem 3 - Applied Calculus

Question

The position \(s\) of a car at time \(t\) is given in the following table. $$\begin{array}{c|c|c|c|c|c|c}\hline t(\mathrm{sec}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\\hline s(\mathrm{ft}) & 0 & 0.5 & 1.8 & 3.8 & 6.5 & 9.6 \\\\\hline\end{array}$$ (a) Find the average velocity over the interval \(0 \leq t \leq\) 0.2. (b) Find the average velocity over the interval \(0.2 \leq t \leq\) 0.4. (c) Use the previous answers to estimate the instantaneous velocity of the car at \(t=0.2\).

Step-by-Step Solution

Answer

(a) 2.5 ft/s, (b) 6.5 ft/s, (c) 4.5 ft/s

Step 1: Understand Average Velocity

Average velocity over a time interval is the change in position divided by the change in time for that interval. The formula is: \( v_{avg} = \frac{s_2 - s_1}{t_2 - t_1} \), where \(s_1\) and \(s_2\) are positions at times \(t_1\) and \(t_2\), respectively.

Step 2: Compute Average Velocity for Interval [0, 0.2)

For the interval [0, 0.2), use \(s_1 = 0\) at \(t_1 = 0\) and \(s_2 = 0.5\) at \(t_2 = 0.2\). The average velocity is: \( v_{avg} = \frac{0.5 - 0}{0.2 - 0} = \frac{0.5}{0.2} = 2.5 \text{ ft/s} \).

Step 3: Compute Average Velocity for Interval [0.2, 0.4)

For the interval [0.2, 0.4), use \(s_1 = 0.5\) at \(t_1 = 0.2\) and \(s_2 = 1.8\) at \(t_2 = 0.4\). The average velocity is: \( v_{avg} = \frac{1.8 - 0.5}{0.4 - 0.2} = \frac{1.3}{0.2} = 6.5 \text{ ft/s} \).

Step 4: Estimate Instantaneous Velocity at t=0.2

The instantaneous velocity at a point can be approximated by the average of the average velocities on either side of the point. Using the results from Steps 2 and 3, estimate the instantaneous velocity at \(t = 0.2\) as the average of 2.5 ft/s and 6.5 ft/s. Thus, the estimated instantaneous velocity is: \( v_{inst} = \frac{2.5 + 6.5}{2} = 4.5 \text{ ft/s} \).

Key Concepts

Instantaneous Velocity

Instantaneous velocity gives us a snapshot of how fast an object is moving at a specific point in time. It differs from average velocity because it focuses on a particular moment. To estimate instantaneous velocity, you look at the average velocities leading up to and immediately following the time point you're interested in.

Here's how we handle it in practice:

• First, determine the average velocities before and after your point of interest.
• Then, take the average of these two velocities to get an approximate value for the instantaneous velocity.

In our example, by averaging the velocities of 2.5 ft/s (before) and 6.5 ft/s (after) at time 0.2 sec, the estimated instantaneous velocity is 4.5 ft/s. This method provides a reasonable estimation of speed at that exact moment.

Position Function

The position function tells us where an object is located at any specific time. In practical terms, it shows how far the object has traveled from its starting point at each given moment.

When interpreting a position function:

• The function's values represent the object's position over time.
• It helps calculate average velocities, using differences in position over specific intervals.

In the original exercise table, the numbers indicate the car's position in feet at intervals of 0.2 seconds. For example, at time 0.4 seconds, the car is 1.8 feet from the starting point. Understanding this function is crucial to analyze how the car's velocity changes over time.

Time Interval Analysis

Time interval analysis involves examining changes over specific periods, helping us understand how an object's velocity evolves. When you know the position of an object at two different times, you can calculate its average velocity over that interval.

Here's how to conduct this analysis:

• Identify the starting and ending positions and times within your interval.
• Apply the formula for average velocity: \( v_{avg} = \frac{\Delta s}{\Delta t} \) (change in position over change in time).
• This gives you the average velocity, providing insights into the object's motion over the interval.

In the example, by assessing the intervals [0, 0.2) and [0.2, 0.4), we calculated average velocities of 2.5 ft/s and 6.5 ft/s, respectively. Such analysis not only helps in estimating instantaneous velocity but also gives a complete picture of how speeds fluctuate within the entire duration.