Toy Gun Spring Compression and Projectile Height: How are they Related?

[dhphysics]

Homework Statement

There are two problems, but I think that they are similar:

A toy gun shoots a projectile straight up. The maximum height reached by the projectile is H when the spring is compressed x cm. For the projectile to reach a height of 2H, the spring of the gun should be compressed how much?The bumper of a car is connected to the body by a spring of spring constant k. When the car hits a cement wall with a speed of 1.0 km/h, the spring compresses 1.0 cm. If the car hits a cement wall with a speed of 2.0 km/h, the spring will compress by how much? (answer in cm)

Homework Equations

The work of the spring is Ws = -.5*k*x^2
The force of the spring is F=-kx

The Attempt at a Solution

Problem 1: If work is force * distance, then the equation of the work of the spring can be set equal to the equation for the force of the spring * the distance H:

-k*x*H = -.5*k*x^2
after some algebra, you get to this equation:
H=1/2x

So if you wanted to double H, you would also double x.

So in terms of x, the spring should be compressed 2x cm. However, this is not the correct answer, but I think my reasoning is correct. Can somebody point out the flaw?

Problem2:
If -.5*k*x^2 = .5*m*v^2,
you would plug in "2*v" for v because 2km/hr is double 1km/hr
Thus, x would be 2 times the original x for 1km/hr, so the spring would compress 2cm.
I'm not sure if this problem is correct, because the previous problem is incorrect so I'm not sure about spring problems.

 

[mfb]

-k*x*H = -.5*k*x^2

Where does the left side come from?
And where did you take gravity into account?
The projectile loses contact to the spring after the first x cm.

The second solution is correct.