A wooden plank of mass `M` and length `L` is floating in still water. A persons of mass `m` starts at one end of the plank and reaches to other end other end in time `t_(0)`, moving with a constant speed. Choose the correct option.
(i) The speed of the person as seen from the ground is smaller than `(L)/(t_(0))`.
(ii) The speed of the plank as seen from the ground is `((m)/(m + M)) (L)/(t_(0))`.
(iii) The speed of the plank as seen from the ground is `((M)/(m + M))(L)/(t_(0))`.
(iv) The total `K.E.` of the system is `(1)/(2) (m + M)((L)/(t_(0)))^(2)`.

Options

A. `(i) ,(ii)`
B. `(i),(iii)`
C. `(i) ,(iii) ,(iv)`
D. `(i),(ii),(iv)`

Answer

A

Explanation

Velocity of the man with respect to the plank is
`u = (L)/(t_(0))`
When the man moves to the left , the plank moves to the right.
Velocity of the plank w.r.t. the ground `= v`
Velocity of the man w.r.t. the ground ` = u - v`
By the momentum conservation
`m(u -v) = Mv rArr v = (mu)/(M + m)`
Velocity of the man w.r.t. the ground `= u -v`
`(Mu)/(M + m) = ((M)/(M + m)) ((L)/(t_(0))) lt ((L)/(t_(0)))`
(i) is `O.K.`
`v = ((m u)/(m + M)) = ((m)/(m + M)) ((L)/(t_(0)))`
(ii) is `O.K.`
`K.E`. of the system `= (1)/(2) Mv^(2) + (1)/(2) m(u - v)^(2)`
`= (1)/(2) ((m M)/(m + M))((L)/(t_(0)))^(2)`
(iv) is wrong.