The speed 1 km/sec is equal to _____ mm/microsecond ?

We're asked to convert #1# #"km/s"# into units of #"mm/"mu"s"#.

We can use dimensional analysis; using conversion factors to convert from one unit to the next and canceling units as we go. The process is as follows:

We'll be converting from kilometers (#"km"#) to millimeters (#"mm"#), and then from seconds (#"s"#) to microseconds (#mu"s"#):

#1(cancel("km"))/(cancel("s"))overbrace(((10^6color(white)(l)"mm")/(1cancel("km"))))^(10^6color(white)(l)"mm in each km")underbrace(((1cancel("s"))/(10^6color(white)(l)mu"s")))_(10^6color(white)(l)mu"s in each s") = color(red)(ulbar(|stackrel(" ")(" "1color(white)(l)"mm/"mu"s"" ")|)#

Therefore, option (C) is the correct answer.

The method I learned for unit conversion is to multiply by 1 in such a way that the existing unit is canceled and the desired unit survives.

The principle is to make use of the fact that #a/b = 1# if #a = b# and that you can multiply anything by 1 without changing its value. Consider the US dollar. #1" dollar" = 100" pennies"#.

Therefore
#(1" dollar") / (100" pennies") = 1# and #(100" pennies")/ (1" dollar") = 1#
Note: There are 2 fractions, both shown to be equal to 1, because you will have to choose which way is needed for the cancellation to work the way you need it to work.

Example:
#5.25 cancel(" dollars") * (100 " pennies")/ (1 cancel(" dollar")) = 525 " pennies"#

This is the conversion from km/s to mm/microseconds:
#(1 cancel("km"))/cancel("s") * (cancel(10^6) " mm")/(1 cancel(" km")) * (1 cancel(" s"))/(cancel(10^6) " microseconds") = 1 " mm"/"microsecond" #

I hope this helps,
Steve