Question

A rain drop has a terminal momentum of 5.0 kgm/s while reaching the ground. If 8 such drops are mixed together, find the terminal momentum of the bigger drop just before reaching the ground ?​

Answer 1

terminal momentum of the bigger drop just before reaching the ground is 160 kgm/s

let radius of smaller rain drop is r and of bigger is R.

volume of bigger drop = 8 × volume of smaller drop

⇒4/3πR³ = 8 × 4/3 πr³

⇒R = 2r .......(1)

also if mass of smaller drop is m then mass of bigger drop = 8 × m = 8m.

as it is given that,

rain drop are moving with terminal velocity. it means, total upward force acting on drop is balanced by weight of it.

i.e., viscous force on smaller drop + upthrust on smaller drop = weight of smaller drop

⇒6πnrv1 + Vρg = mg .......(1)

similarly for bigger drop,

viscous force on bigger drop + upthrust on bigger drop = weight of bigger drop

⇒6πn(2r)v2 + 8Vρg = 8mg ........(2)

from equations (1) And (2) we get,

v2 = 4v1

given, terminal momentum of smaller drop = 5kgm/s

so, mv1 = 5kgm/s ......(3)

then terminal momentum of bigger drop = (8m)(v2)

= 8m × 4v1 = 32mv1

from equation (3),

= 32 × 5 = 160 Kgm/s

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Answer 2

$\huge\bold\purple{Answer:-}$

let radius of smaller rain drop is r and of bigger is R.

volume of bigger drop = 8 × volume of smaller drop

⇒4/3πR³ = 8 × 4/3 πr³

⇒R = 2r .......(1)

also if mass of smaller drop is m then mass of bigger drop = 8 × m = 8m.

as it is given that,

rain drop are moving with terminal velocity. it means, total upward force acting on drop is balanced by weight of it.

i.e., viscous force on smaller drop + upthrust on smaller drop = weight of smaller drop

⇒6πnrv1 + Vρg = mg .......(1)

similarly for bigger drop,

viscous force on bigger drop + upthrust on bigger drop = weight of bigger drop

⇒6πn(2r)v2 + 8Vρg = 8mg ........(2)

from equations (1) And (2) we get,

v2 = 4v1

given, terminal momentum of smaller drop = 5kgm/s

so, mv1 = 5kgm/s ......(3)

then terminal momentum of bigger drop = (8m)(v2)

= 8m × 4v1 = 32mv1

from equation (3),

= 32 × 5 = 160 Kgm/s