Question

The United States has high-speed trains running between Boston and New York capable of speeds up to \(160 \mathrm{mi} / \mathrm{h}\). Are these trains faster or slower than the fastest trains in the United Kingdom, which reach speeds of \(225 \mathrm{~km} / \mathrm{h} ?\)

Short answer

The high-speed trains in the United States, with a converted speed of \(257.4944 \frac{km}{h}\), are faster than the United Kingdom's fastest trains, which reach speeds of \(225 \frac{km}{h}\).

Step-by-step solution

Conversion factor from miles to kilometers

To convert the high-speed train's velocity in the United States from mi/h to km/h, we need to know the conversion factor between miles and kilometers. There are approximately 1.60934 kilometers in 1 mile.

Convert US train speed to km/h

To convert the speed from mi/h to km/h, we must multiply the speed given in mi/h by the conversion factor obtained in step 1: \( Speed_{US} = 160 \frac{mi}{h} \times 1.60934 \frac{km}{mi} \) The miles unit cancel out, and we get the speed in km/h: \( Speed_{US} = 257.4944 \frac{km}{h} \)

Compare the speeds of the US and UK trains

Now that we have both speeds in km/h, we can compare them: Speed of US trains: \(257.4944 \frac{km}{h}\) Speed of UK trains: \(225 \frac{km}{h}\) Since \(257.4944 > 225\), the United States' high-speed trains are faster than the United Kingdom's fastest trains.

Key concepts

Unit Conversion

Understanding unit conversion is essential in many scientific and everyday contexts. It involves translating a quantity expressed in one set of units to another, maintaining the same magnitude of that quantity. In applications like comparing speeds, distances, or temperatures from different systems of measurement, it's crucial to use unit conversion to make accurate and meaningful comparisons.

For example, to compare the speed of trains in the United States (measured in miles per hour, or mph) and the United Kingdom (measured in kilometers per hour, or km/h), a conversion factor is used. This factor is a numerical value that allows for the direct conversion of one unit into another. In this case, the conversion factor from miles to kilometers is approximately 1.60934. By multiplying the speed in miles per hour by this factor, we convert it to kilometers per hour, allowing us to accurately compare the speeds irrespective of the units used.

Speed Conversion

Speed conversion specifically applies unit conversion principles to the measurement of velocity. It's critical in fields where different speed measurement systems interact, such as international travel, global logistics, and scientific research that spans across countries using different units. When converting speed, it's important to ensure that the conversion factor is applied correctly to avoid errors in calculation and to ensure accurate comparisons.

To convert from miles per hour to kilometers per hour like in the original problem, multiply the given speed by the factor of 1.60934. For the reverse process, divide the speed in kilometers per hour by the same factor. Proper speed conversion not only gives a numeric result but also changes the unit appropriately, as seen when calculating the speed of US high-speed trains into km/h for a direct comparison with UK train speeds.

Mathematics in Chemistry

Mathematics plays a pivotal role in chemistry, helping to quantify and describe physical properties, reactions, and processes. It allows chemists to establish stoichiometric relationships, calculate reaction yields, determine concentration levels, and understand the kinetics and thermodynamics of reactions.

Unit conversion is especially valuable in chemistry, which frequently requires switching between various units to measure quantities such as mass, volume, concentration, and pressure. For instance, converting between moles and grams involves the molar mass as a conversion factor. Understanding and applying the correct conversion factors ensures accurate measurements and comparisons, which are the backbone of valid chemical experimentation and research.