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  4. A body of mass M is dropped from a height h on a sand floor. If the body penetrates x cm into the sand, the average resistance offered by the sand to the body is

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Q. A body of mass $M$ is dropped from a height $h$ on a sand floor. If the body penetrates $x\, cm$ into the sand, the average resistance offered by the sand to the body is

Work, Energy and Power

Solution:

the body strikes the sand floor with a velocity $v$, then
Potential energy = Kinetic energy
$Mgh=\frac{1}{2}M\upsilon^{2}$
With this velocity $v$, when body passes through the sand floor it comes to rest after travelling a distance $x$. Let $F$ be the resisting force acting on the body. Net force in downward direction
$= Mg - F$
Work done by all the forces is equal to change in KE
$\left(Mg-F\right)x=0-\frac{1}{2}M\upsilon^{2}$
$\left(Mg-F\right)x=-Mgh$
or $Fx=Mgh+Mgx$
or $F=Mg\left(1+\frac{h}{x}\right)$