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Question

Two spherical rain drops with radii in the ratio 1:2 fall from a great height through the atmosphere. The ratio of their momenta after they have attained terminal velocity is:

A
1:8
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B
2:1
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C
1:32
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D
1:2
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Solution

The correct option is C 1:32
Given, the ratio of radii of the drops, r1:r2=1:2
The terminal velocity and mass of a body is:
vT=29r2g(σρ)η
where σ is the density of the body i.e rain drop and ρ is density of fluid (air) through which the body is falling.
vTr2 ...(i)
Similarly, mass of rain drop will be,
m=σ×43πr3
mr3 ...(ii)
From Eq.(i) & (ii),
mvTr5
Hence, when rain drops attain vT, the ratio of their momenta will be:
p1p2=m1v1m2v2=(r1r2)5
p1p2=(12)5
p1:p2=1:32

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