Question

The winds in a hurricane can reach almost 200 miles per hour. What is this speed in meters per second? (Assume three significant figures.)

Short answer

The speed of the hurricane in meters per second is approximately 89.4 m/s.

Step-by-step solution

Convert miles to meters

To convert miles to meters, use the conversion factor 1 mile = 1609.34 meters. Multiply the speed of the hurricane in miles per hour by this conversion factor.

Convert hours to seconds

There are 3600 seconds in one hour. To convert the speed from meters per hour to meters per second, divide by 3600.

Calculate the speed in meters per second

Multiply the speed in miles per hour by the conversion factor for miles to meters, then divide by the number of seconds in an hour to get the speed in meters per second. Perform the calculation to three significant figures.

Key concepts

Significant Figures

When we are dealing with measurements in chemistry and physics, it's not just the value that matters, but also the precision of that value. This precision is communicated through the use of significant figures. Significant figures, or 'sig figs', as they are often abbreviated, are the digits in a number that carry meaning contributing to its measurement accuracy.

For example, in the hurricane speed mentioned in the exercise, the speed is given as 'almost 200 miles per hour' and we are instructed to assume three significant figures. This means that the number '200' is considered to have three digits that are significant for our calculation, namely the '2' and two implied zeros that follow it. When we convert this value to another unit, care must be taken to ensure that we do not imply a greater precision than is present in the original measurement. Once the conversion is performed, the result should also be expressed with three significant figures to maintain consistency in the precision of our data.

Conversion Factors

In unit conversions, a conversion factor is a multiplier that allows you to change a quantity expressed in one set of units to an equivalent quantity in another set of units. The conversion factor is constructed so that the old unit cancels out, leaving the new unit.

A conversion factor can be based on any two quantities known to be equivalent. For instance, to convert miles to meters, as needed to solve the hurricane wind speed problem, we use the conversion factor 1 mile = 1609.34 meters. It's crucial that the conversion factor represents a known equivalent quantity, which means it should not affect the accuracy of the measurement, other than changing the units. This is why it's expressed with as many significant figures as the known precision of that relationship.

Speed Conversion

Converting a speed measurement from one set of units to another often utilizes both of the concepts we've already discussed: conversion factors and significant figures. Speed conversion is not just a simple unit conversion because it involves two units that need to be converted simultaneously: distance and time.

In the hurricane example, the speed needs to be converted from miles per hour to meters per second. This involves two steps: first, converting the distance from miles to meters using the conversion factor for miles to meters; and second, converting the time from hours to seconds. There are exactly 3600 seconds in one hour, and this is a defined value with an infinite number of significant figures because it is exact. Hence, the conversion from hours to seconds does not limit the number of significant figures in our final answer. The entire process needs to be handled while considering the significant figures from the original measurement.

Dimensional Analysis

The best way to approach unit conversions is through a method called dimensional analysis. This technique involves using the units of measurement as a guide in your calculations, treating them almost like algebraic quantities. It’s a systematic way to use conversion factors to change from one unit to another.

For instance, in converting the speed of a hurricane from miles per hour to meters per second, dimensional analysis involves writing the given value with its unit, then systematically multiplying by conversion factors such that the units you want to get rid of (miles, hours) are canceled out, leaving only the units you want to end with (meters, seconds). Each step of the conversion is clear and shows exactly how units are being converted, which also helps to minimize errors and make sure that the calculations are done correctly.