Problem 56 - Calculus Single Variable

Question

Speed measurements of a runner taken at half-second intervals during the first 5 seconds of a sprint are provided in the following table.$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Time (s) } & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 & 4.5 & 5.0 \\\\\hline \text { Speed (m/s) } & 0 & 5.26 & 6.67 & 7.41 & 8.33 & 8.33 & 9.52 & 9.52 & 10.64 & 10.64 & 10.87 \\\\\hline\end{array}$$ About how many meters did the athlete run during that 5-second interval? Use Simpson's Rule.

Step-by-Step Solution

Answer

The runner traveled approximately 40.97 meters during the first 5 seconds.

Step 1: Understanding the Problem

We need to calculate the approximate distance traveled by a runner during the first 5 seconds of a sprint. The speed values at intervals of 0.5 seconds are given. Simpson's Rule will be used to approximate the integral of speed over time, which represents the distance traveled.

Step 2: Applying Simpson's Rule Formula

Simpson's Rule for approximating the integral of a function is given by:\[\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]\]where \(\Delta x\) is the width of each subinterval, \(n\) is an even integer, and \(f(x_i)\) denotes the speed values.

Step 3: Setting Parameters for Calculation

The interval is from 0 to 5 seconds, divided into 10 equal parts, making \(n = 10\). The width \(\Delta x\) is 0.5 seconds. The speed values given correspond to \(f(x_0), f(x_1), \ldots, f(x_{10})\).

Step 4: Calculating Using Simpson's Rule

Substitute the given speed values and \(\Delta x = 0.5\) into Simpson's Rule:\[\text{Distance} = \frac{0.5}{3} \left[ 0 + 4 \times 5.26 + 2 \times 6.67 + 4 \times 7.41 + 2 \times 8.33 + 4 \times 8.33 + 2 \times 9.52 + 4 \times 9.52 + 2 \times 10.64 + 4 \times 10.64 + 10.87 \right]\]

Step 5: Simplifying and Evaluating the Expression

Compute each part and sum them:\[= \frac{0.5}{3} \left[ 0 + 21.04 + 13.34 + 29.64 + 16.66 + 33.32 + 19.04 + 38.08 + 21.28 + 42.56 + 10.87 \right]\]After summing:\[= \frac{0.5}{3} \times 245.83\]\[= 0.1667 \times 245.83 \approx 40.97 \, \text{meters}\]

Step 6: Concluding the Calculation

According to Simpson's Rule, the runner traveled approximately 40.97 meters during the first 5 seconds of the sprint.

Key Concepts

Numerical Integration

When we talk about numerical integration, we're diving into methods used to calculate the integral of a function when an exact formula is difficult or impossible to utilize. This is particularly useful when we have a set of discrete data points instead of a continuous function. Numerical methods come into play to approximate the area under a curve, which in many real-world scenarios, like in physics and engineering, represent quantities such as total distance, area, or volume.

One of the well-known methods in numerical integration is **Simpson's Rule**. This rule provides an efficient way to get an approximation of the integral by using quadratic polynomials. Unlike other methods that might use simpler shapes like rectangles or trapezoids, Simpson's Rule achieves higher accuracy by using parabolas to approximate the segments of the curve. This approach specifically calculates the integral by applying polynomial functions to subintervals of the domain.

• For Simpson’s Rule, the interval needs to be divided into an even number of subintervals.
• The area under the curve in each subinterval is approximated using a quadratic function that fits through three equidistant points.
• Once areas for each segment are computed, they are summed to estimate the total integral.

Understanding numerical integration through methods like Simpson’s Rule allows one to practically apply calculus principles to evaluate integrals that arise in potential real-world situations.

Distance Calculation

In physics, the concept of distance plays a core role in understanding motion. When determining how far an object has traveled within a given time frame, we often look to measure the area under a speed-time graph. This area provides a direct representation of the distance covered over the time interval.

If perfect and continuous speed data isn't available, calculating the distance moves into the realm of approximation, often using numerical methods. Yields from these methods approximate the integral of speed over time, producing the distance:

• For a constantly changing speed, a graph plotting speed against time might look quite complex.
• By using recorded speeds at specific intervals, a piecewise representation of the curve is constructed.
• Then, Simpson's Rule or other numerical integration techniques are leveraged to calculate the area under this reconstructed curve.

This approach was used in computing the runner's distance over five seconds, allowing the approximation from speed data to deliver actionable insights about performance.

Speed Measurements

Speed measurements describe how quickly an object covers distance over time. It's a vector quantity, meaning it doesn't just measure "how fast" but also "in which direction" the object is moving. In the exercise, speed values were recorded at regular half-second intervals for the runner, transforming the data into something valuable for analysis.

Understanding the role of speed Measurements in computational and realistic scenarios is pivotal:

• By recording speeds at consistent time intervals, it is possible to map the changes in velocity over the duration of motion.
• Insights derived from these measurements allow for deeper exploration into acceleration, total travel distance, and performance over the sprint.
• This type of analysis is crucial in sports science, vehicle performance evaluation, and robotics.

Collecting accurate speed measurements and applying appropriate computational methods allows for a detailed understanding of motion dynamics, feeding into more extensive analytical and strategic evaluations.