Problem 35 - Thomas Calculus

Question

Thoroughbred racing A racehorse is running a 10 -furlong race. (A furlong is 220 yards, although we will use furlongs and seconds as our units in this exercise.) As the horse passes each furlong marker \((F),\) a steward records the time elapsed \((t)\) since the beginning of the race, as shown in the table: $$ \begin{array}{lllllllll}{F} & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline t & {0} & {20} & {33} & {46} & {59} & {73} & {86} & {100} & {112} & {124} & {135}\end{array} $$ a. How long does it take the horse to finish the race? b. What is the average speed of the horse over the first 5 furlongs? c. What is the approximate speed of the horse as it passes the 3 -furlong marker? d. During which portion of the race is the horse running the fastest? e. During which portion of the race is the horse accelerating the fastest?

Step-by-Step Solution

Answer

a. 135 sec, b. ~0.0685 furlongs/sec, c. ~0.0769 furlongs/sec, d. 2-3 & 3-4 furlongs, e. 0-1 furlong.

Step 1: Calculate Total Time

To determine how long it takes the horse to finish the race, we look at the time elapsed at the 10th furlong marker. From the table, when the horse reaches the 10th furlong, the time is 135 seconds.

Step 2: Calculate Average Speed for First 5 Furlongs

The average speed is defined as total distance divided by total time. Over the first 5 furlongs, the horse covers a distance of 5 furlongs in a time of 73 seconds. Thus, the average speed is \( \frac{5\,\text{furlongs}}{73\,\text{seconds}} \approx 0.0685\,\text{furlongs per second} \).

Step 3: Estimate Speed at 3 Furlong Marker

To approximate the speed at the 3-furlong marker, we can compute the average speed between 2 furlongs and 4 furlongs, assuming constant speed between these points. Between 2 and 3 furlongs, the times are 33 and 46 seconds (a difference of 13 seconds for 1 furlong), and between 3 and 4 furlongs, 46 and 59 seconds (also a 13 second difference for 1 furlong). The approximate speed at the 3-furlong mark is thus \( 1\,\text{furlong} / 13\,\text{seconds} \approx 0.0769\,\text{furlongs per second} \).

Step 4: Determine Fastest Running Portion

To find the fastest portion, look for the smallest time interval over 1 furlong. Checking the intervals: 2-3 furlongs (13 seconds), 3-4 (13 seconds), and others are larger, the horse runs the fastest between 2 and 3 furlongs and 3 and 4 furlongs, as both have the shortest duration of 13 seconds.

Step 5: Determine Fastest Accelerating Portion

Acceleration is the rate of change of speed. Estimate acceleration by comparing speed differences over consecutive furlongs. From 1-2 (20-33), 2-3 (33-46), observe where speeds change most. Acceleration is highest where speed difference is greatest between each consecutive furlong. The transition from 0-1 furlong to 1-2 furlongs shows the initial speed increase, suggesting highest acceleration there.

Key Concepts

Average Speed

Average speed is a fundamental concept in calculus, representing the total distance traveled divided by the total time taken. In simple terms, it's like asking, "On average, how fast was something moving?" for a particular journey.

To calculate average speed, use the formula:

• Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} \)

For example, in the thoroughbred racing scenario, the horse's average speed over the first 5 furlongs is calculated by dividing the 5 furlongs traveled by 73 seconds, resulting in approximately 0.0685 furlongs per second.

Average speed helps us understand the overall efficiency of motion over a period and doesn't account for any speed variations within that period.

Acceleration

Acceleration in calculus refers to how quickly an object’s speed changes over time. It measures how speed increases or decreases, and it can be thought of as the "speed of the speed". You can determine acceleration by calculating the change in speed over a known time interval.

In the racehorse example, you would look at changes in speed over consecutive furlongs. For instance, comparing the speed from 0 to 1 furlongs and from 1 to 2 furlongs can show where the change is most significant.
The section where speed shows the most substantial change characterizes the portion of the race where acceleration is at its peak. In this context, the greatest acceleration is noted early in the race.

Acceleration is crucial for understanding how quickly a horse can increase its speed, which is particularly important in races.

Distance-Time Analysis

Distance-time analysis is a tool used to understand the motion of an object by examining its location over time. This analysis is essential in calculus for visualizing motion patterns and assessing speed changes.

The given table with furlongs and time points allows you to analyze how the horse's speed varies throughout the race. By focusing on the differences in time for each interval between furlongs, we can identify where the horse runs faster or slower.

This analysis is valuable for spotting where the horse might speed up or slow down, helping coaches and trainers make strategic decisions in competitive settings.

• Quick intervals indicate faster speeds.
• Longer intervals hint at a decrease in speed.

Instantaneous Speed

Instantaneous speed is the rate at which an object is moving at a specific moment in time. Unlike average speed, it provides a snapshot of motion at a particular point.

In the race example, to find the horse's speed at the 3-furlong marker, you examine short intervals before and after this point. By calculating the average speed between 2 to 3 furlongs and 3 to 4 furlongs (both resulted in approximately 0.0769 furlongs per second), we can approximate the instantaneous speed around the 3-furlong mark.

Instantaneous speed is essential for understanding precise speed at any moment, which is critical for performance analysis and strategy adjustments during a race.