Problem 25 - Calculus

Question

Find the average velocity over the interval \(0 \leq t \leq 0.2\) and estimate the velocity at \(t=0.2\) of a car whose position, \(s\), is given by the following table. $$\begin{array}{c|cccccc}\hline t \text { (sec) } & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\\hline s(\mathrm{ft}) & 0 & 0.5 & 1.8 & 3.8 & 6.5 & 9.6 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Average velocity is 2.5 ft/s, and estimated velocity at \(t=0.2\) is 6.5 ft/s.

Step 1: Understanding the Problem

We need to find the average velocity of the car over the time interval from \( t = 0 \) to \( t = 0.2 \) seconds. The position of the car is given by the table, where \( s(t) \) represents the position in feet at time \( t \) in seconds.

Step 2: Formula for Average Velocity

The average velocity, \( v_{avg} \), over a time interval \( [a, b] \) is calculated using the formula:\[v_{avg} = \frac{s(b) - s(a)}{b - a}\]Here, \( a = 0 \), \( b = 0.2 \), \( s(a) = s(0) = 0 \), and \( s(b) = s(0.2) = 0.5 \).

Step 3: Calculate Average Velocity

Substitute the values into the formula:\[v_{avg} = \frac{0.5 - 0}{0.2 - 0} = \frac{0.5}{0.2} = 2.5 \text{ feet per second}\]Thus, the average velocity of the car over the interval \(0 \leq t \leq 0.2\) is 2.5 feet per second.

Step 4: Estimating Instantaneous Velocity at \(t = 0.2\)

To estimate the velocity at \( t = 0.2 \), calculate the average velocity over a tiny interval around \( t = 0.2 \). Consider the interval \( [0.2, 0.4] \) with \( s(0.2) = 0.5 \) and \( s(0.4) = 1.8 \).

Step 5: Calculate Velocity Estimate at \(t = 0.2\)

Calculate the velocity over the interval \( [0.2, 0.4] \):\[v = \frac{s(0.4) - s(0.2)}{0.4 - 0.2} = \frac{1.8 - 0.5}{0.2} = \frac{1.3}{0.2} = 6.5 \text{ feet per second}\]This value gives an estimate of the velocity at \( t = 0.2 \).

Key Concepts

Understanding the Position Function

The position function is how you describe movement in terms of a mathematical relationship between position and time. Here, we are given a position function through a table that shows the car's position at various time intervals.

• The position function is represented by the variable \( s(t) \), which denotes the car's position in feet at time \( t \) in seconds.
• In a typical scenario, the position function can be given as an equation, but here it's in table form showing discrete position values at specific times.

When solving problems like this, it's crucial to interpret the position data correctly. We need to use this position information to compute different types of velocities by examining how the position changes over time.
In our example, as time progresses (from \( t = 0 \) to \( t = 1.0 \) seconds), the car's position increases, giving us the data we need to calculate velocities.

Estimating Instantaneous Velocity

Instantaneous velocity refers to the speed and direction of an object at a particular instant. Since we cannot directly determine the instantaneous velocity when we only have discrete position data, we estimate it.

• To approximate the instantaneous velocity at \( t = 0.2 \), we use a small interval around that point.
• This involves calculating the difference in position over a narrow time interval to approximate what would be the slope of the tangent line to the position-time curve at \( t = 0.2 \).

In this exercise, we considered the interval from \( t = 0.2 \) to \( t = 0.4 \). By using limits so small, the average velocity over this small range gives a good approximation of the instantaneous velocity at \( t = 0.2 \).
This method, essentially, is about zooming in on the graph and capturing the car's motion at a specific split-second, embodying the concept of instantaneous velocity.

Using Interval Calculation for Average Velocity

To find how fast something is going over a specific time period, we use interval calculations. Average velocity is a basic form of this, where the performance over an entire interval is averaged out.

• The average velocity is calculated over the interval \([a, b]\) using the formula: \( v_{avg} = \frac{s(b) - s(a)}{b - a} \).
• In our task, to find the average velocity from \( t = 0 \) to \( t = 0.2 \), position has changed from 0 to 0.5 feet.
• Plugging in the values, we find \( v_{avg} = \frac{0.5 - 0}{0.2} = 2.5 \, \text{feet per second} \).

Interval calculations help understand not just instantaneous happenings, but trends over time, which is a key reason why average velocities are often easier to work with for practical purposes.