A bullet enters in a piece of wood with velocity `v_(0)` and the resistive force acting on the bullet in the vood is proportional to `v^((1)/(3))`. If the total distance travelled by the bullet is proportional to `(v_(0))^(beta)`, then the value of `beta` is

Options

A. `(2)/(3)`
B. `(5)/(3)`
C. `(4)/(3)`
D. `-(1)/(3)`

Answer

B

Explanation

Resistive force `= -kv^((1)/(3))`
So, retardation of bullet is
`a = (dv)/(dt) = -(k)/(m) v^((1)/(3)) rArr (dv)/(dx).(dx)/(dt) = - (k)/(m) v^((1)/(3))`
`rArr (dv)/(dx).v = -(k)/(m). V^((1)/(3)) rArr v^(3) dv = - (k)/(m) dx`
Integrating with limits, we get (s = total displacement)
`underset(v_(0))overset(0)int v^((2)/(3)) dv = - (k)/(m) underset(0)overset(s)int dx`
`[(3)/(5) v^((5)/(3))]_(v_(0))^(0) = - (k)/(m) [x]_(0)^(s) rArr - (3)/(5) v_(0)^((5)/(3)) = - (k.s)/(m)`
or `s prop v^((5)/(3))`