#### Answer

The rocket should be launched when the horizontal distance is 7.62 meters.

#### Work Step by Step

We can find the vertical acceleration of the rocket.
$\sum F = ma$
$F_{thrust}-mg=ma$
$a = \frac{F_{thrust}-mg}{m}$
$a = \frac{8.0~N-(0.500~kg)(9.80~m/s^2)}{0.500~kg}$
$a = 6.2~m/s^2$
We can find the time it takes for the rocket to reach a height of 20 meters.
$y = \frac{1}{2}at^2$
$t = \sqrt{\frac{2y}{a}}$
$t = \sqrt{\frac{(2)(20~m)}{6.2~m/s^2}}$
$t = 2.54~s$
We can find the horizontal distance the rocket travels in this time. Note that the rocket's horizontal speed will be equal to the cart's speed.
$x = v_x~t$
$x = (3.0~m/s)(2.54~s)$
$x = 7.62~m$
The rocket should be launched when the horizontal distance is 7.62 meters.