A small spherical ball of radius 0.1 mm and density 10⁴ kg m⁻³ falls freely under gravity through a distance h before entering a tank of water. If after entering the water the velocity of the ball does not change and it continues to fall with the same constant velocity inside the water, then the va…

Question

A small spherical ball of radius 0.1 mm and density 10⁴ kg m⁻³ falls freely under gravity through a distance h before entering a tank of water. If after entering the water the velocity of the ball does not change and it continues to fall with the same constant velocity inside the water, then the value of h will be m. (Given g=10 m s⁻², viscosity of water =1.0×10⁻⁵ N s m⁻²)

Accepted Answer

20. Given: r=0.1 mm=10⁻⁴ m, ρ_b=10⁴ and ρ_w=10³ kg m⁻³, η=10⁻⁵ N s m⁻², g=10 m s⁻². Idea: the speed being unchanged on entry means the ball hits the surface already moving at the terminal velocity for water. vₜ=(2r²(ρ_b-ρ_w)g)/(9η)=(2×10⁻⁸×(9×10³)×10)/(9×10⁻⁵)=(1.8×10⁻³)/(9×10⁻⁵)=20 m s⁻¹. The fall through air is free, so vₜ²=2gh. h=(vₜ²)/(2g)=(400)/(20)=20 m.

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