A body of mass m= 1kg is moving in a medium and experiences a fractions force `F= -kv`, where v is the speed of the body. The initial speed is `v_(0)= 10 ms^(-1)` and after 10s, its energy becomes half of initial energy. Then, the valeu of k is

Options

A. 10 ln `sqrt2`
B. ln `sqrt2`
C. `("ln" 2)/(20)`
D. 10 ln 2

Answer

C

Explanation

`because` According to the equation, `(1)/(2) mv_(f)^(2) = (1)/(2) xx (1)/(2) mv_(i)^(2)` (`v_(i), v_(f)`= initial and final speeds of the body)
or `v_(f)^(2) = (v_(i)^(2))/(2) or v_(f) = (10)/(sqrt2)`
Given, `f= -kx`
or `ma= -kv rArr (mdv)/(dt)= -kv`
`rArr underset(10)overset(10//sqrt2)int (1)/(v) dv= - underset(0)overset(10)int kdt (because m= 1kg)`
`rArr (ln v)_(10)^(10//sqrt2)= -k (10)`
`rArr "ln" (10)/(sqrt2)- ln 10= -k(10)`
`rArr k= (1)/(10)ln ((10)/(10//sqrt2))= (1)/(10) ln sqrt2= (ln 2)/(20)`