A sphere is dropped under gravity through a fluid of viscosity `eta`. Taking the average ac celeration as half of the initial ac celeration, show that the time to attain the terminal velocity is independent of the fluid density.

Explanation

To solve the problem of a sphere dropped under gravity through a fluid of viscosity \(\eta\) and show that the time to attain terminal velocity is independent of the fluid density, we can follow these steps:

### Step 1: Identify the forces acting on the sphere
When a sphere is dropped in a viscous fluid, three forces act on it:
1. The gravitational force (\(F_g\)): \(F_g = mg\), where \(m\) is the mass of the sphere and \(g\) is the acceleration due to gravity.
2. The buoyant force (\(F_b\)): \(F_b = V \rho_f g\), where \(V\) is the volume of the sphere and \(\rho_f\) is the density of the fluid.
3. The drag force (\(F_d\)): According to Stokes' law, for a sphere moving through a viscous fluid, \(F_d = 6 \pi \eta r v\), where \(\eta\) is the viscosity of the fluid, \(r\) is the radius of the sphere, and \(v\) is the velocity of the sphere.

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