Question

The speed limit on many U.S. highways is \(65 \mathrm{mi} / \mathrm{hr}\). Convert this speed into each alternative unit. (a) \(\mathrm{km} /\) day (b) \(\mathrm{ft} / \mathrm{s}\) (c) \(\mathrm{m} / \mathrm{s}\) (d) \(\mathrm{yd} / \mathrm{min}\)

Short answer

The speed limit of 65 mi/hr converts to approximately (a) 2517.66 km/day, (b) 95.33 ft/s, (c) 29.06 m/s, (d) 1920 yd/min.

Step-by-step solution

Convert miles to kilometers

To convert the speed from miles per hour to kilometers per day, use the conversion factor that 1 mile is approximately 1.60934 kilometers. Since there are 24 hours in a day, multiply 65 miles/hour by 1.60934 and then by 24 to convert to kilometers/day.

Convert miles to feet and hours to seconds

To convert the speed from miles per hour to feet per second, use the conversion factor that 1 mile equals 5280 feet and 1 hour equals 3600 seconds. Divide 65 miles/hour by 3600 to convert to miles/second, and then multiply by 5280 to convert to feet/second.

Convert miles to meters and hours to seconds

To convert the speed from miles per hour to meters per second, use the conversion factor that 1 mile is approximately 1609.34 meters and 1 hour equals 3600 seconds. Divide 65 miles/hour by 3600 to convert to miles/second, and then multiply by 1609.34 to convert to meters/second.

Convert miles to yards and hours to minutes

To convert the speed from miles per hour to yards per minute, utilize the conversion factor that 1 mile equals 1760 yards and 1 hour equals 60 minutes. Multiply 65 miles/hour by 1760 to convert miles to yards, and then divide by 60 to convert hours to minutes.

Key concepts

Dimensional Analysis

Dimensional analysis is a fundamental method used in science, engineering, and mathematics to convert one set of units to another, maintaining the same physical quantity. This process involves various conversion factors which are scientifically determined ratios that express the same quantity in different units.

For example, in the speed conversion exercise, dimensional analysis is utilized to change the unit of speed from miles per hour to other units such as kilometers per day or feet per second. Each step within the conversion must preserve the dimension of speed, which is distance over time. The correct conversion factors adjust the distance and time units independently to achieve the final desired unit, ensuring an accurate conversion while maintaining the same physical representation of speed.

Conversion Factors

Conversion factors play a crucial role in the process of unit conversion. They are the tools that enable you to convert from one unit to another through multiplication or division. A conversion factor is a ratio that is always equal to one, because it represents the same quantity expressed in two different measurement units.

For instance, in the provided solution, various conversion factors are applied. The factor of 1 mile equals 1.60934 kilometers is used for spatial conversion, while temporal conversion relies on the fact that there are 24 hours in a day, 60 minutes in an hour, and 3600 seconds in an hour. By using these factors, the original speed given in miles per hour can be accurately converted to kilometers per day, feet per second, meters per second, or yards per minute.

Rate Conversions

Rate conversions involve changing the units of rates, which are typically expressed as a ratio of two different units, such as distance per time for speed. The correct application of rate conversions is necessary to ensure the numerical value accurately represents the same rate after changing the units.

Through the step-by-step solution, the speed limit is converted into various units by carefully modifying both the numerator (distance) and the denominator (time), ensuring the speed 'rate' remains constant. For example, converting miles per hour to meters per second requires both converting the distance from miles to meters and the time from hours to seconds. The goal is to make sure the rate reflects the same speed, regardless of the units used, and understanding the relationship between different units of distance and time is key for accurate rate conversions.

Application of Conversion Factors in Rate Conversions

The real challenge lies in recognizing which conversion factors to use and how to apply them. As seen in the exercise, it requires a combination of the conversion factors for distance and time. This two-pronged approach is a practical example of how rates are effectively converted from one unit to another while ensuring the rate's integrity, which is critical for applications such as navigating speed limits in different countries or converting technical specifications in scientific research.