Answer
The rocket should be launched when the horizontal distance is 38.9 meters.
Work Step by Step
We can find the acceleration of the rocket.
$\sum F = ma$
$F_{thrust}-mg=ma$
$a = \frac{F_{thrust}-mg}{m}$
$a = \frac{15.0~N-(0.800~kg)(9.80~m/s^2)}{0.800~kg}$
$a = 8.95~m/s^2$
We can find the time it takes for the rocket to reach a height of 30 meters.
$y = \frac{1}{2}at^2$
$t = \sqrt{\frac{2y}{a}}$
$t = \sqrt{\frac{(2)(30~m)}{8.95~m/s^2}}$
$t = 2.59~s$
We can find the distance the target travels in this time.
$x = v~t$
$x = (15~m/s)(2.59~s)$
$x = 38.9~m$
The rocket should be launched when the horizontal distance is 38.9 meters.
