An archer stands 40.0m from the target. if the arrow is shot horizontally with a velocity of 90.0 m/s, how far above the bull's eye must she aim to compensate for gravity pulling her arrow downward? ... ?

Ok, first we need to have in mind that to findthe distance we need to find also the time then we use the kinematic equations in the x-direction

delta x = Vox*t + 1/2*a*t^2

since there is no horizontal acceleration, a = 0 m/s/s

delta x = Vox*t

40.0 m = 90 m/s * t

t = 4/9 s

Now using this time you can calculate the vertical displacement due to gravity.

delta y = Voy*t + 1/2*g*t^2

since there is no initial vertical velocity, Voy = 0m/s

delta y = 1/2*g*t^2

delta y = 1/2 * (-9.81 m/s/s) * (4/9 s) ^2

solve for delta y and you will have the answer