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## Homework Statement

To find the stiffness of a material, a block of that material is placed 30 cm under a metallic cone with 0.3 kg; the cone is free falling from rest, penetrating a distance

*x*in the block until it stops. It is known that when the cone penetrates in the block the force of the block exerced on the cone is

*kx*where

^{2}*k*is a constant that depends on the stiffness to penetration of the material; if the cone penetrates a distance

*x*= 5 cm, find the value of the constant

*k*.

An added restriction that I'm adding to the problem is to not use the relations between Work and Energy (at least directly) -> that subject wasn't approached yet.

## Homework Equations

The derivatives of x(t) and v(t), and:

[tex]a = v\frac{dv}{dx}[/tex]

[tex]F = m a[/tex]

## The Attempt at a Solution

According to the solutions and to the Work-Energy relations, k should be 24696 N/m

^{2}. But I've tried several times and the values don't match:

[tex]F - W = m a[/tex]

[tex]k {x}^{2} - 9.8 m = m a[/tex]

[tex]k {x}^{2} - m 9.8 = m v \frac{dv}{dx}[/tex]

[tex]\int_{0.05+0.3}^{0}k {x}^{2} - m 9.8\,dx = \frac{1}{2} ({0}^{2}-{0}^{2})[/tex]

[tex]-\frac{343 k - 24696}{24000} = 0[/tex]

[tex]k = 72[/tex]