The speed of light in a vacuum is $ 2.998 \times 10^8$ m/sec. What is this speed in miles per hour?

Question

Peace to all. Studying and trying to get a handle on how to calculate like-minded questions, I came across this question from the web. I don't understand how exactly it was solved. How were the units canceled out and with what exactly?

Solution:
To convert this measurement, we need to convert meters to miles and seconds to hours. To do this, we need the following relationships: 1000 meter = 1 kilometer; 1 kilometers = 0.621 mile; 60 seconds = 1 minute; 60 minutes = 1 hour; We can now set up the equation using these relationships so the units cancel out leaving only the desired miles/hour.

speedMPH = 2.998 x 10^8 m/sec x (1 km/1000 m) x (0.621 mi/1 km) x (60 sec/1 min) x (60 min/1 hr)

Note all the units canceled out, leaving only miles/hr: > > speedMPH = (2.998 x 10^8 x 1/1000 x 0.621 x 60 x 60) miles/hr > > speedMPH = 6.702 x 10^8 miles/hr

Answer:
The speed of light in miles per hour is 6.702 x 10^8 miles/hr.

Answer 1

Rather than tackling the problem at hand, here is a simpler example. Suppose we start with a speed of 60 mph and want to convert to miles per minute. Since 1 hour is 60 minutes, we have $$60\text{ mph} = \frac{60\text{ miles}}{1\text{ hour}} = \frac{60\text{ miles}}{60\text{ min}} = 1\text{ mile/min}.$$ Note that we have replaced 1 hour by 60 minutes 'by hand'. This is effective for a single unit conversion, but seems tedious if multiple conversions had to be done. To make things more practical, we instead rewrite the above computation as such:

$$60\text{ mph} = \frac{60\text{ miles}}{1\text{ hour}}\times\frac{1 \text{ hour}}{60 \text{ minutes}} = 1\text{ mile/min}.$$

Note that we've exchanged "replace 1 hour by 60 minutes" with "multiply by 1 hour / 60 minutes." The reason this works is because 1 hour = 60 minutes, so (1 hour)/(60 minutes) is just equal to one. So we multiplied 60 mph by 1, leaving the quantity itself unchanged; however, the old units cancel out in the process and thereby carry out unit conversion.

In this example, of course, the problem is simple enough that it doesn't make a difference. But suppose we wanted to convert 60 mph into units of feet per second. Since one mile is 5280 feet, one hour is 60 minutes, and one minute is 60 seconds, we can carry out all three unit conversions by the following multiplication:

\begin{align} 60\text{ mph} &= \frac{60\text{ miles}}{1\text{ hour}} \times \frac{1\text{ hour}}{60\text{ minutes}} \times \frac{1\text{ minute}}{60\text{ seconds}} \times \frac{5280\text{ feet}}{1\text{ mile}} \\ &= \frac{60\times 5280}{60\times 60} \frac{\text{feet}}{\text{second}}\\ & = 88\text{ ft/sec}. \end{align}

Note how the multiplications are arranged to cancel out hours, minutes, and miles.

Can you see how to apply this idea to your problem?

Answer 2

Notice how some units cancel when the proper terms are multiplied.

$$1\space \frac{m}{s}\approx 2.237\space mph \\ \implies 299792458 \space \frac{m}{s} \space \times\space 2.237\space \frac{mph}{m} \space\times\space 1\space h \space \times\space 3600\space \frac{s}{h}\\ = \,2,414,288,622,766\space mph$$