Be sure the units in the unit conversion cancel correctly. If the unit conversion factor was written upside down, the units do not cancel correctly in the equation. We see the “miles” in the numerator in 0.50 mi/min cancels the “mile” in the denominator in the first conversion factor. Also, the “min” in the denominator in 0.50 mi/min cancels the “min” in the numerator in the second conversion factor.
Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of meters per second and, after the cancellations, the only units left are a meter (m) in the numerator and a second (s) in the denominator, so we have indeed obtained these units.
Check Your Understanding Light travels about 9 Pm in a year. Given that a year is about
what is the speed of light in meters per second?
The density of iron is
under standard conditions. Convert this to kg/m
3 .
Strategy
We need to convert grams to kilograms and cubic centimeters to cubic meters. The conversion factors we need are
and
However, we are dealing with cubic centimeters
so we have to use the second conversion factor three times (that is, we need to cube it). The idea is still to multiply by the conversion factors in such a way that they cancel the units we want to get rid of and introduce the units we want to keep.
Solution
Significance
Remember, it’s always important to check the answer.
Be sure to cancel the units in the unit conversion correctly. We see that the gram (“g”) in the numerator in 7.86 g/cm
3 cancels the “g” in the denominator in the first conversion factor. Also, the three factors of “cm” in the denominator in 7.86 g/cm
3 cancel with the three factors of “cm” in the numerator that we get by cubing the second conversion factor.
Check that the units of the final answer are the desired units. The problem asked for us to convert to kilograms per cubic meter. After the cancellations just described, we see the only units we have left are “kg” in the numerator and three factors of “m” in the denominator (that is, one factor of “m” cubed, or “m
3 ”). Therefore, the units on the final answer are correct.
Check Your Understanding We know from
[link] that the diameter of Earth is on the order of 10
7 m, so the order of magnitude of its surface area is 10
14 m
2 . What is that in square kilometers (that is, km
2 )? (Try doing this both by converting 10
7 m to km and then squaring it and then by converting 10
14 m
2 directly to square kilometers. You should get the same answer both ways.)
Unit conversions may not seem very interesting, but not doing them can be costly. One famous example of this situation was seen with the
Mars Climate Orbiter . This probe was launched by NASA on December 11, 1998. On September 23, 1999, while attempting to guide the probe into its planned orbit around Mars, NASA lost contact with it. Subsequent investigations showed a piece of software called SM_FORCES (or “small forces”) was recording thruster performance data in the English units of pound-seconds (lb-s). However, other pieces of software that used these values for course corrections expected them to be recorded in the SI units of newton-seconds (N-s), as dictated in the software interface protocols. This error caused the probe to follow a very different trajectory from what NASA thought it was following, which most likely caused the probe either to burn up in the Martian atmosphere or to shoot out into space. This failure to pay attention to unit conversions cost hundreds of millions of dollars, not to mention all the time invested by the scientists and engineers who worked on the project.
