A cone falling with a speed `v_(0)` strikes and penetrates the block of a packing material. The acceleration of the cone after impact is
`a=g-cx^(2)`. Where c is a positive constant and x is the penetration distance. If maximum penetration depth is `x_(m)` then c equals

Options

A. `(2gx_(m)+v_(0)^(2))/(x_(m)^(2))`
B. `(2gx_(m)-v_(0)^(2))/(x_(m)^(2))`
C. `(6gx_(m)-3v_(0)^(2))/(2x_(m)^(3))`
D. `(6gx_(m)+3v_(0)^(2))/(2x_(m)^(3))`

Answer

D

Explanation

`v(dv)/(dx)=g-cx^(2)`
`impliesint_(v_(0))^(0)vdv=gint_(0)^(x)dx-cint_(0)^(x)x^(2)dx`
`implies-(v_(0)^(2))/(2)=gx-(cx^(3))/(3)`
`implies(x^(3)c)/(3)=gx+(v_(0)^(2))/(2)impliesc=(3g)/(x^(2))+(3v_(0)^(2))/(2x^(3))`