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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^{2} where 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 WorkEnergy 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]
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