Which of the following statements are true ?
(i) A uniform wooden plank of mass `150 kg` and length `8 m` is floating on still water with a man of `50 kg` at one end of it. The man walks to the other end of the plank and stops. The distance covered by the plank is `2 m`.
(ii) A kid of mass `15 kg` is sitting in a boat of mass `45 kg` in a lake. The distance of kid from the bank of lake is `12 m`. Now the body moves inside the boat at a distance `4m` towards the bank and stops. The distance of the kid from the bank (there is no friction between the water and the boat ) is `11 m`.
(iii) Two persons of masses `40 kg` and `60 kg` are sitting at the midpoint of a `12 -m`- long boat `(140 kg)` standing still in water. Now they move to opposite ends of the boat. Neglecting the friction between boat and water, the distance traveled by the boat in water is `50 cm`.
(iv) In a gravity free space , a man of mass `m` standing at a height `h` above the floor, throws a ball of mass `m_(0)` straight down with a speed `v_(0)`. When the ball reaches the floor, the distance of the man above the floor is `(1 + (m_(0))/(m))h`.

Options

A. `(i),(ii)`
B. (i) ,(iii) ,(iv)`
C. `(ii) ,(iii) ,(iv)`
D. all of the above

Answer

B

Explanation

In the following cases , `vec(F)_(ext) = 0` , the system is initially at rest and hence the location of center of mass is fixed.
(i) Let the distance covered by the plank is `d`.
`m_(1) x_(1) = m_(2) x_(2)`
`50 (8 - d) = 150 d rArr d = 2m`
`(i) is O.K.`
(ii) Let the distance travelled by boat is `d`.
`15(4 - d) = 45 d rArr d = 1m`
Distance of boy from the bank of the river is
`(12 - 4) + d = 8 + 1 = 9 m`
`(ii)` is incorrect.
(iii) Let the distance travelled by the boat be `d`.
`60(6 -d) = 40( 6 + d) + 140 d`
`rArr d = (1)/(2) m = 50 cm`
`(iii) is O.K`.
(iv) `m_(0) h = mh' rArr h' = (m_(0))/(m) h`
Distance of man above the floor is
`h + h' = (1 + (m_(0))/(m))h`
`(iv) is O.K`.