Problem 31 - Calculus: Early Transcendental Functions

Question

The table shows the velocity of a projectile at various times. Estimate the distance traveled. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { time (s) } & 0 & 0.25 & 0.5 & 0.75 & 1.0 & 1.25 & 1.5 & 1.75 & 2.0 \\ \hline \text { velocity (ft/s) } & 120 & 116 & 113 & 110 & 108 & 106 & 104 & 103 & 102 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Based on the detailed calculations, the estimated total distance traveled will be the summation of the calculated distances for each time interval. Since the exact calculations depend on the application of the trapezoidal rule, the exact figure may vary slightly based on the numeric calculations.

Step 1: Identify the time intervals and corresponding velocities

The time intervals are given as [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1.0], [1.0, 1.25], [1.25, 1.5], [1.5, 1.75], and [1.75, 2.0]. The corresponding velocities for these intervals are [120, 116], [116, 113], [113, 110], [110, 108], [108, 106], [106, 104], [104, 103] and [103, 102].

Step 2: Estimate the area for each time interval

Let's estimate the area (and thus, the distance traveled) for each time interval using the formula for the area of a trapezoid: \(A = 0.5 * (b1 + b2) * h\), where \(b1\) and \(b2\) are the velocities at each endpoint of the interval and \(h\) is the duration of the time interval. It is important to note that the units of measurement should be consistent for this formula to work.

Step 3: Calculate distances for each time interval

Now, calculate the distance traveled over each time interval. The distance over a given time interval will be the product of the average velocity over that interval and the time duration. The average velocity is the average of the initial and final velocity over that time interval.

Step 4: Sum up all the distances

The total distance traveled by the projectile is the sum of the distances traveled over each time interval. Sum up the distances calculated in the previous step to estimate the total distance traveled.

Key Concepts

Trapezoidal Rule

Understanding the trapezoidal rule is essential when we need to estimate the area under a curve, which is often represented by data points in practical applications like calculating distance from velocity-time data. Imagine plotting the velocity against time on a graph – you would get a shape that resembles a series of trapezoids side by side.

The trapezoidal rule estimates the area of each trapezoid by taking the average of the two bases, which in our case are the velocities at the start and end of each time interval, and multiplying by the height, which is the duration of the time interval. Mathematically, it’s expressed as: \[ A = \frac{1}{2} \times (b1 + b2) \times h \] where \( b1 \) and \( b2 \) are the bases, and \( h \) is the height of the trapezoid.

By using the trapezoidal rule to find these areas and summing them up, we can estimate the total distance traveled by the projectile between the initial and final times provided.

Practical Use in Projectile Motion

Specifically, when looking at projectile motion problems like the one in our example, the trapezoidal rule becomes a pragmatic approach to integrate the velocity function and find the approximate value of displacement or distance. It strikes a balance between simplicity and accuracy, especially when dealing with linear or near-linear segments between the data points.

Velocity-Time Data

Velocity-time data is a set of measurements that shows how the velocity of an object changes over time. In the context of projectile motion, this data reflects how the projectile's speed varies as it moves.

Each pair of data points in our example represents the projectile's velocity at specific times, forming a discrete set. The data helps us understand the motion's nature and is crucial for performing calculations that estimate the distance traveled, such as using the trapezoidal rule.

Analyzing The Velocity Change

Velocity-time data may indicate a certain pattern; for instance, a linear decrease in velocity could suggest constant deceleration. By plotting these data points, one can visualize the projectile's behavior over time and make informed guesses about the forces at play, such as air resistance or gravity.

Distance Estimation

Distance estimation in calculus often leans on the integral of a function over a specified interval. In terms of motion, it refers to the process of determining how far an object has traveled based on velocity over time. Our problem requires estimating the distance a projectile travels, given its changing velocity over equal time intervals.

We integrate the velocity function across the time of interest to find the distance. Because we don’t have a continuous function but a series of discrete data points, we use numerical methods like the trapezoidal rule to approximate the distance.

Converting Data to Distance

Each estimated area under the curve (or trapezoidal area) between the time intervals corresponds to an estimated piece of the total distance. Summing these estimates provides a reasonable approximation of the total distance traveled by the projectile, which is powerful in predicting real-world scenarios without needing complex calculations.

Average Velocity

Average velocity, which is central to estimating distance in projectile motion, is defined as the total displacement divided by the total time taken. However, when dealing with variable velocity as in our example, average velocity can also be understood as the mean value of the starting and ending velocities over a small time interval.

Using the average velocity offers a simplified but effective method to compute the distance traveled during each interval by multiplying it by the time elapsed. For the projectile motion problem provided, the average velocity for any given interval can be calculated as: \[ \text{Average Velocity} = \frac{b1 + b2}{2} \] where \( b1 \) and \( b2 \) represent the initial and final velocities of the time interval, respectively.

Importance in Distance Calculations

This approach smoothens out the variations in velocity over small intervals, making the math more manageable and providing a fair approximation for the distance covered in each segment without having to solve a continuous integral.