A cone of mass m falls from a height h and penetrates into sand. The resistance force R of the sand is given by `R=kx^2`. If the cone penetrates up to a distance `x=d` where `dltlamda`, then find the value of k.

Explanation

Force cone while it is penetrating the sand is shown in F.B.D below Applying work energy theorem to the cone as x changes from 0 to d
`triangleK=`work done by `mg+` work done by resistive force R
`F_(final)-K(initial)=mgd-int_0^dkx^2dx`
`0-mg=mgd-int_0^dkx^2dx`
`(kd^3)/(3)=(mgd+gh)impliesk=(3mg)/(d^3)(h+d)`