Yes, you are on the right track. The equation you have written is the correct use of the conservation of mechanical energy. Let's now solve it step-by-step.
The equation you stated is:
1/2mv^2 + mgh = 1/2mv^2 + mgh + 1/2kx^2
Where:
m = mass of the object (2.0 kg)
v = initial velocity (3.0 m/s)
g = acceleration due to gravity (9.8 m/s^2, assuming the object is near the surface of the Earth)
h = height from the horizontal plane (0, because the object is moving the plane horizontally)
k = spring constant (800 N/m)
x = spring displacement
Now let's simplify the equation step-by-step:
1. Cancel out the common terms:
1/2mv^2 = 1/2mv^2
2. Cancel out the masses as well:
0 + mgh = 0 + mgh + 1/2kx^2
3. Remove the redundant zero on the left side:
mgh = mgh + 1/2kx^2
4. Rearrange the equation to solve for x:
mgh - mgh = 1/2kx^2
5. Cancel out the common term on the left side:
0 = 1/2kx^2
6. Solve for x:
1/2kx^2 = 0
Since the coefficient on the right side is zero, we can conclude that x (the spring compression) is also zero. This means that the spring has not been compressed at all when the object comes to rest.