Experimentally it has been found that the force F needed to compress elastically a ball through a distance x (as shown in the figure) follows the formula F ( x ) = ax + bx 2 + cx 3 where a, b and c ar

Question

Experimentally it has been found that the force F needed to compress elastically a ball through a distance x (as shown in the figure) follows the formula F ( x ) = ax + bx 2 + cx 3 where a, b and c are constants. The small ball of mass m at rest is dropped from a great height h. It bounces elastically off the floor and is compressed a maximum distance 'd' during the bounce. The height h is :- (where 'g' is the acceleration due to gravity)

TestHub · Accepted Answer

Elastic bounce: gravitational PE converts fully into the compression energy:

$m g h = \int_{0}^{d} (a x + b x^{2} + c x^{3}) d x = \frac{a d ^{2}}{2} + \frac{b d ^{3}}{3} + \frac{c d ^{4}}{4}$

$h = \frac{1}{m g} (\frac{a d ^{2}}{2} + \frac{b d ^{3}}{3} + \frac{c d ^{4}}{4})$

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