## Related questions with answers

In a test to determine the crushing characteristics of polystyrene packing material, a steel cone of mass $m$ is dropped so that it falls a distance $h$ and then penetrates the material. The resistance $R$ of polystyrene to penetration depends on the cross-sectional area of the penetrating object and thus is proportional to the square of the cone penetration distance $x$, or $R=k x^2$. If the cone comes to rest at a distance $x=d$, determine the constant $k$ in terms of the test conditions and results.

Solution

VerifiedWe begin the solution by writing the known variables of the problem. $\\$ The resistance $R$ is given as:

$R = kx^2$

$k$ just being a constant of proportionality. $\\$ We also know the velocity of the cone at the moment of penetration because it's falling in a gravity field and the velocity of impact is known to be:

$v_f = \sqrt{2gh}$

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