Answer:
In a vertical settling tank, particles settle due to the difference in velocities between the settling velocity (us) and the overflow velocity (v0). The settling velocity is the velocity at which particles settle under the influence of gravity, while the overflow velocity is the velocity at which the liquid overflows from the tank.

Importance of Settling Velocity and Overflow Velocity:
The settling velocity and overflow velocity play a crucial role in determining whether a particle will settle in a vertical settling tank or not. Let's understand the options given in the question and why option 'C' is the correct answer.

Option A: (us = u0)
If the settling velocity is equal to the overflow velocity, it means that the particle is settling at the same rate as the liquid is overflowing. In this case, the particle will not settle in the tank as it will be carried away by the overflowing liquid. Hence, option A is incorrect.

Option B: (us ≤ u0)
If the settling velocity is less than or equal to the overflow velocity, it means that the particle is settling at a slower rate than the liquid is overflowing. In this scenario, the particle will not be able to settle completely and will be carried away by the overflowing liquid. Therefore, option B is also incorrect.

Option C: (us ≥ u0)
If the settling velocity is greater than or equal to the overflow velocity, it means that the particle is settling at a faster rate than the liquid is overflowing. In this case, the particle will be able to settle in the vertical settling tank as it can overcome the overflow velocity and settle at the bottom of the tank. Hence, option C is the correct answer.

Option D: Settling does not depend on velocity
This option is incorrect as settling does depend on velocity. Settling is the process by which particles settle under the influence of gravity, and the settling velocity plays a crucial role in determining the settling behavior of particles in a vertical settling tank. Hence, option D is incorrect.

Conclusion:
In a vertical settling tank, particles will settle when the settling velocity (us) is greater than or equal to the overflow velocity (v0). This allows the particles to settle at the bottom of the tank, overcoming the overflow velocity.