A rocket on a cart is fired to go through a hoop

[beccabrummer]

Homework Statement

A 300 g rocket is on a cart that is rolling to the right at a speed of 2 m/s. The rocket engine, when it is fired, exerts an 6 N thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 18 m above the launch point. At what horizontal distance left of the hoop should you launch?

Homework Equations

F=ma
delta x= V0t+1/2(a)(t^2)
Vf^2=V0^2+2ax
Vf=V0+at

The Attempt at a Solution

I tired taking the force from thrust and subracting the force due to gravity and finding the force of the rocket (3.06N). I then used that to find the acceleration and used that to find the time it takes to reach 18 m vertical. From there I used the time and the velocity of the cart to find the horizontal distance of the cart. Its not the correct answer but I got 7.06m. Need help to discover where I went wrong.

 

[scottdave]

The 3.06 Newtons (net force) looks correct, but what is the net acceleration that you calculated? Can you show us what was the time that you calculated, and how you got it?
I am going to assume that you got 3.53 seconds (from your answer of 7.06m) but how did you arrive at that?