Length in seconds of a crotchet (quarter note USA)

Somewhere on sheet music, it should tell you how long a musical note is worth. It usually shows you how long a crochet (quarter note in the USA) is. 

Here is an example:

In the above song,

#quarter   =   120

tells you that there are 120 crotchets (or quarter notes) in 60 seconds.

From this, it is easy to work out how long one crotchet (or quarter note) is by dividing 60 seconds by 120 notes:

#quarter   = 120 in 60 seconds

#quarter   = `60/120` = 0.5 seconds per #quarter

One crotchet (or quarter note) = 0.5 seconds

Why don't the people who write the sheet music just put on the sheet that a crotchet (or quarter note) is half a second in the first place?

This is because metronomes can be set to that 120 figure. On a metronome, they call this 'beats per minute' (BPM).

To practice playing to sheet music, it can help to play to a metronome.

The interval between one click and another on the metronome represents the length of a crotchet (or quarter note) in this case.