Question

Secretariat is known as the horse with the fastest run in the Kentucky Derby. If Secretariat's record \(1.25-\mathrm{mi}\) run lasted J minute \(59.2\) seconds, what was his average speed in \(\mathrm{m} / \mathrm{s}\) ?

Short answer

Secretariat's average speed during his record-breaking Kentucky Derby run was approximately \(16.87 \ \frac{\text{m}}{\text{s}}\).

Step-by-step solution

Convert distance from miles to meters

To convert 1.25 miles to meters, we will use the conversion factor 1 mile = 1609.34 meters. Multiplying the given distance by the conversion factor, we have: \(1.25 \ \text{miles} \times 1609.34 \ \frac{\text{meters}}{\text{mile}} = 2011.675\ \text{meters}\)

Convert time from minutes and seconds to seconds

We are given the time taken as 1 minute 59.2 seconds. First, we will convert minutes to seconds by multiplying with 60 (since there are 60 seconds in a minute): \(1 \ \text{minute} = 1 \times 60 \ \text{seconds} = 60 \ \text{seconds}\) Now, add the converted minutes and seconds together: \(60 \ \text{seconds} + 59.2 \ \text{seconds} = 119.2 \ \text{seconds}\)

Calculate the average speed

Now that we have the distance covered in meters and the time taken in seconds, we can calculate the average speed as follows: \(speed = \frac{distance}{time}\) Substituting the values, we have: \(speed = \frac{2011.675 \ \text{meters}}{119.2 \ \text{seconds}} ≈ 16.87 \ \frac{\text{m}}{\text{s}}\) The average speed of Secretariat during his record-breaking Kentucky Derby run was approximately 16.87 m/s.

Key concepts

Unit Conversion

Unit conversion is crucial in solving problems that involve different measurement systems. For Secretariat's run, the distance was given in miles, but we needed it in meters to calculate the speed in meters per second (m/s). This is often the case in physics and engineering problems where consistent units are essential.

To convert miles to meters, we use the conversion factor 1 mile = 1609.34 meters. This means that when you multiply the number of miles by 1609.34, you get the distance in meters. For example, for a distance of 1.25 miles, we perform the following calculation:

• Multiply: 1.25 miles × 1609.34 m/mile = 2011.675 meters.

This gives us the distance in the correct unit to proceed with the speed calculation.

Distance and Time

Understanding how to appropriately measure and convert distance and time is foundational in physics. In the given problem, the distance Secretariat covered was already converted to meters. Now, the focus is on converting time.

We were given a time of 1 minute and 59.2 seconds. To find the total time in seconds, we convert the minute into seconds, since 1 minute is equal to 60 seconds. Here's how we do it:

• Convert the minute: 1 min = 1 × 60 = 60 seconds.
• Add to the given seconds: 60 seconds + 59.2 seconds = 119.2 seconds.

This total of 119.2 seconds is now ready to be used to calculate the average speed of Secretariat's incredible run.

Speed Formula

The speed formula is a simple yet powerful tool for calculating how fast something is moving. Speed is defined as the distance traveled divided by the time it takes to travel that distance. The formula is expressed as:

\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]

To find Secretariat's average speed, we use the distance we converted to meters, and the time we converted to seconds. Plugging these values into the speed formula provides us with the answer:

• Distance = 2011.675 meters.
• Time = 119.2 seconds.
• Speed = \( \frac{2011.675}{119.2} \approx 16.87 \; \text{m/s} \).

This calculation tells us that Secretariat ran at an incredible average speed of approximately 16.87 meters per second during the Kentucky Derby.