Idon't get it at all! speed is one thing; distance is another. where is the arrow you shoot up at 50m/s when it runs out of speed?

... Gravity makes it go 9.8 m/s slower each second.
    So it runs out of speed in (50/9.8) = 5.1 seconds after the shot.

... At the beginning of the shot, its speed is 50 m/s.
    When it runs out of speed, its speed is zero.
    Its AVERAGE speed during the whole 5.1 seconds is  25 m/s .

... Going straight up for 5.1 seconds, at an average speed of  25 m/s,
     it rises to an altitude of

                               (25 m/s) x (5.1 sec)  =  127.5 meters  .

(You said you have an answer of 125 meters.
Whoever worked out that one probably used 10 m/s²
for the acceleration of gravity, instead of the 9.8 that I used.) 

Of course, once the arrow "runs out of speed", it starts falling,
faster and faster, until it hits the ground.  You could calculate
how long it takes to fall, and how fast it's going when it hits the
ground.  But all of that is for another question.