bluewood
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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 kx2 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:
a = v\frac{dv}{dx}
F = m a
The Attempt at a Solution
According to the solutions and to the Work-Energy relations, k should be 24696 N/m2. But I've tried several times and the values don't match:
F - W = m a
k {x}^{2} - 9.8 m = m a
k {x}^{2} - m 9.8 = m v \frac{dv}{dx}
\int_{0.05+0.3}^{0}k {x}^{2} - m 9.8\,dx = \frac{1}{2} ({0}^{2}-{0}^{2})
-\frac{343 k - 24696}{24000} = 0
k = 72