Problem 115 - Chemistry

Question

Distances over land are measured in statute miles ( \(5280 \mathrm{ft}\) ), but distances over water are measured in nautical miles, where 1 nautical mile was originally defined as 1 minute of arc along an Earth meridian, or \(1 / 21,600\) of the Earth's circumference through the poles. A ship's speed through the water is measured in \(k\) nots, where 1 knot \(=1\) nautical mile per hour. Historically, the unit knot derived from the practice of measuring a ship's speed by throwing a log tied to a knotted line over the side. The line had a knot tied in it at intervals of \(47 \mathrm{ft} .3\) in., and the number of knots run out in 28 seconds was counted to determine speed. (a) How many feet are in a nautical mile? How many meters? (b) The northern bluefin tuna can weigh up to 1500 pounds and can swim at speeds up to 48 miles per hour. How fast is this in knots? (c) A league is defined as 3 nautical miles. The Mariana Trench, with a depth of 35,798 feet, is the deepest point in the ocean. How many leagues deep is this? (d) By international agreement, the nautical mile is now defined as exactly 1852 meters. By what percentage does this current definition differ from the original definition, and by what percentage does it differ from a statute mile?

Step-by-Step Solution

Answer

(a) Nautical mile: 6080 feet, 1853.25 meters. (b) 41.711 knots. (c) 1.963 leagues. (d) 0.0675% difference from historical nautical mile, 15.08% from statute mile.

Step 1: Define a Nautical Mile in Feet

Historically, a nautical mile is defined as 1/21,600 of the Earth's circumference through the poles. Given the Earth's circumference is approximately 24,901 miles, we convert this to feet: 24,901 miles * 5280 feet/mile = 131,473,280 feet. Dividing by 21,600, one nautical mile = 6,080 feet.

Step 2: Convert Feet to Meters

Using the conversion factor 1 meter = 3.28084 feet, we convert 6,080 feet to meters: 6,080 feet * (1 meter/3.28084 feet) ≈ 1,853.25 meters.

Step 3: Convert Speed from Miles to Knots

The northern bluefin tuna swims at 48 miles per hour. Since 1 statute mile = 5,280 feet and 1 nautical mile = 6,080 feet, 1 mile is approximately 0.868976 knots. Therefore, 48 miles/hour * 0.868976 knots/mile = 41.711 knots.

Step 4: Calculate Leagues from Feet

A league is defined as 3 nautical miles, and from Step 1, 1 nautical mile = 6,080 feet. Therefore, 1 league = 3 * 6,080 = 18,240 feet. The depth of the Mariana Trench is 35,798 feet. So, 35,798 feet / 18,240 feet/league ≈ 1.963 leagues.

Step 5: Calculate Percentage Difference for Nautical Mile in Meters

The current nautical mile is precisely 1,852 meters. From Step 2, the historical definition in meters is about 1,853.25. Therefore, the percentage difference is ((1,853.25 - 1,852) / 1,852) * 100 ≈ 0.0675%.

Step 6: Calculate Percentage Difference from Statute Mile

Using the current nautical mile definition (1,852 meters), convert the statute mile to meters using 1 mile = 1,609.34 meters. The percentage difference is ((1,852 - 1,609.34) / 1,609.34) * 100 ≈ 15.08%.

Key Concepts

Conversion of Units

Understanding conversion of units is crucial, particularly when working with nautical measurements. A nautical mile differs from a land-based statute mile, reflecting the specific needs of marine navigation. Converting between these units requires precision. For example, to convert nautical miles to feet, you use the formula: 1 nautical mile = 6,080 feet. When converting miles to knots, a conversion factor of approximately 0.868976 is used. This means that a speed in statute miles per hour can be converted to knots by multiplying by this factor. For distance conversions, where 1 meter equals approximately 3.28084 feet, we find the nautically used modern definition of a nautical mile as exactly 1,852 meters. Mastering these conversions allows for effective and accurate cross-reference between measurements in different contexts.

Speed Measurement

Speed measurement at sea is an essential skill, and the unit of knots is historically intriguing. A knot measures how fast a vessel moves through the water: 1 knot equals 1 nautical mile per hour. This unit originated from an old sailing practice where sailors would measure speed by letting out a line over the side of the ship. The line had knots tied at regular intervals of approximately 47 feet 3 inches. By counting how many knots ran out in a set period of 28 seconds, sailors could estimate their speed. In modern contexts, converting from speeds given in statute miles per hour to knots involves using the conversion factor 0.868976. For instance, a northern bluefin tuna swimming at 48 mph translates to about 41.711 knots.

Historical Definitions

Nautical terms often have fascinating historical roots. Initially, a nautical mile was defined as one minute of arc along any meridian of the Earth; this was equivalent to 1/21,600 of the Earth's circumference. At that time, the circumference was roughly estimated to be 24,901 miles. Using this old definition, one nautical mile translated to 6,080 feet. Over time, this definition underwent refinements to suit international standardization, resulting in the current definition of exactly 1,852 meters. This historical evolution underscores the nautical mile's profound significance in aligning global maritime navigation standards.

Depth Measurement

Depth measurement is critical in understanding oceanography and involves its own set of units. One such unit is the league, which is equivalent to three nautical miles. Converting this to feet, using 1 nautical mile = 6,080 feet, we find that 1 league equals 18,240 feet. This is particularly useful when measuring extreme ocean depths. For instance, the Mariana Trench, known as the deepest point in the ocean, is 35,798 feet deep. To express this depth in leagues, you divide 35,798 by 18,240, giving an approximate depth of 1.963 leagues. Using leagues for such measurements provides a practical perspective by conveying depth in a scale more relatable than raw feet.