Dimensional analysis<\/strong> is a systematic approach to solving complex unit conversion problems. It allows us to keep track of our units and ensure that they cancel out correctly, giving us confidence in the accuracy of our final answer. While this approach might seem tedious for straightforward problems, it proves to be exceptionally useful when dealing with more complex conversions involving multiple steps and units.<\/p>\r\n Dimensional analysis<\/strong> is a method used to convert from one unit to another. It involves multiplying the original measurement by one or more conversion factors until we convert the initial unit to the desired unit.<\/p>\r\n<\/div>\r\n<\/section>\r\n Units are essential in measurements to express quantity. We utilize various units in daily life, such as meters for distance, kilograms for mass, and seconds for time. When working with quantities measured in different units, it's often necessary to convert from one unit to another. The conversion factor is the tool we use to accomplish this. A conversion factor<\/strong> is a ratio expressing how many of one unit are equal to another unit. In the sections for U.S. units of measurement and metric systems you were given conversion factors to switch between units. These conversion factors were then used in the factor label method to convert between units. The process is similar in the dimensional analysis of rates. When converting a rate, we multiply by a conversion factor.<\/p>\r\n To correctly use dimensional analysis, you must identify the appropriate conversion factor based on the units involved. To determine which conversion factors are needed, it's essential to identify the starting unit (the unit you have) and the final unit (the unit you want to convert to).<\/p>\r\n Consider a scenario where you want to convert kilometers to meters. The initial unit is kilometers, and the final unit is meters. The conversion factor that relates kilometers and meters is [latex]1[\/latex] km [latex]= 1000[\/latex] m. Sometimes, you might need to use multiple conversion factors.<\/p>\r\n For example, suppose you have a speed of [latex]50[\/latex] miles per hour (mph), and you want to convert this speed to feet per second (fps). This conversion requires two conversion factors:<\/p>\r\n Once you've identified the appropriate conversion factor(s), you can use it to convert rates from one unit to another. This process involves multiplying the rate you want to convert by the conversion factor(s), thus changing the units but preserving the quantity's value.<\/p>\r\n You have likely encountered distance, rate, and time problems in the past. This is likely because they are easy to visualize and most of us have experienced them firsthand. In our next examples, we will solve distance, rate, and time problems that will require us to change the units that the distance or time is measured in. Let's go back to our previous example and solve for the speed in feet per second.<\/p>\r\n Next, we can set up the conversion:<\/p>\r\n [latex]50 \\frac{\\text{miles}}{\\text{hour}} \\times (\\frac{5280 \\text{ feet}}{1 \\text{ mile}}) \\times (\\frac{1 \\text{ hour}}{3600 \\text{ seconds}})[\/latex]<\/p>\r\n Notice how the units you are converting from (miles and hours) cancel out, leaving you with the units you want (feet and seconds).<\/p>\r\n Doing the math gives a result of about [latex]73.33[\/latex] feet per second, which is the speed converted from mph to fps.<\/p>\r\n [\/hidden-answer]<\/p>\r\n<\/section>\r\n To answer this question, we need to convert [latex]20[\/latex] seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don\u2019t, we will need to do additional unit conversions. We will need to know that [latex]5280[\/latex] ft [latex]= 1[\/latex] mile. We might start by converting the [latex]20[\/latex] seconds into hours:<\/p>\r\n [latex]\\displaystyle{20}\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{1}{180}\\text{ hour}[\/latex]<\/p>\r\n Now we can multiply by the [latex]15[\/latex] miles\/hr:<\/p>\r\n [latex]\\displaystyle\\frac{1}{180}\\text{ hour}\\cdot\\frac{15\\text{ miles}}{1\\text{ hour}}=\\frac{1}{12}\\text{ mile}[\/latex]<\/p>\r\n Now we can convert to feet:<\/p>\r\n [latex]\\displaystyle\\frac{1}{12}\\text{ mile}\\cdot\\frac{5280\\text{ feet}}{1\\text{ mile}}=440\\text{ feet}[\/latex]<\/p>\r\n We could have also done this entire calculation in one long set of products:<\/p>\r\n [latex]\\displaystyle20\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{15\\text{ miles}}{1\\text{ miles}}=\\frac{5280\\text{ feet}}{1\\text{ mile}}=\\frac{1}{180}\\text{ hour}[\/latex]<\/p>\r\ndimensional analysis<\/h3>\r\n
Determining Conversion Factors<\/h3>\r\n
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Converting a Rate Using Conversion Factors<\/h3>\r\n
\r\n[hidden-answer a=\"439949\"]We have already determined which conversion factors we need to convert the speed:\r\n\r\n\r\n\t
\r\n[reveal-answer q=\"946318\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"946318\"]\r\n\r\n
\r\nView the following video to see this problem worked through.<\/p>\r\n