How Far Will the Spring Compress When a Block is Dropped?

[jedjj]

Homework Statement

A block of mass m = 2.0 kg is dropped from height h = 55 cm onto a spring of spring constant k = 1960 N/m. Find the maximum distance the spring is compressed.

Homework Equations

$$\Delta K+ \Delta U_G+ \Delta U_S=0$$ \change in kinetic energy+change in gravitational energy+change in spring energy=0
$$\Delta K=0$$
$$U_{Gf}-U_{Gi}+U_{Sf}-U_{Si}=0$$
$$U_{Si}=0$$ can be assumed
so
$$U_{Sf}=U_{Gi}-U_{Gf}$$

The Attempt at a Solution

resumed from above:
$$\frac {kx_f^2}{2}=mgy_i-mgy_f$$
so: $$\frac {kx_f^2}{2}=mgy_i-mgx_f$$
With this I have tried over and over to solve for $$x_f$$ but I cannot find any way of getting $$x_f$$ by itself.
edit:LaTeX formatting

 

[Antineutron]

conservation of energy, $$\Delta$$U(g)+ $$\Delta$$U(spring) = 0 <=== look, all the energy lost from one object goes to the other, so the sum is 0. Remember! conservation of energy. Think about it.

$$\Delta$$U(g)=-$$\Delta$$U(spring) <===== moved that equation around a it... just algebra.

mgh2-mgh1 = (1/2)kx^2 <===== look what we have here if we make it look more detailed with what we know.

Or simply mgh=-(1/2)kx^2

 

[hage567]

$$\frac {kx_f^2}{2}=mgy_i-mgx_f$$
With this I have tried over and over to solve for $$x_f$$ but I cannot find any way of getting $$x_f$$ by itself.

It's a quadratic equation.