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**Hint:**If the bucket is moving in a vertical circle with constant speed then it is performing vertical circular motion. Vertical circular motion is non-uniform circular motion. Know the concept of non-uniform circular motion and the formula for the speed and how it varies at different points on the circle.

**Complete step by step solution:**

When a particle or body follows a circular path at varying speed then it is said to perform non-uniform circular motion. As the speed of the body keeps changing thus there is a presence of tangential acceleration along with the radial acceleration in non-uniform circular motion. When a body moves in a vertical circular motion, the speed of the body at the bottom and at the top of the circle is not equal and thus as the speed of the body changes in vertical circular motion, it is non-uniform circular motion. Velocity of the body varies at different points in the circle.

Velocities of the body at different position are given as:

At the lowermost point ${v_L} = \sqrt {5gr} $ where $g$ is acceleration due to gravity and $r$ is the radius of the circle

At the horizontal or middle point ${v_M} = \sqrt {3gr} $

At the top most point of the circle ${v_T} = \sqrt {gr} $

Now that we know the formula for the velocity at the top most point of the circle, we can find the speed such that the water in the bucket does not spill when the bucket is at the highest point by substituting the values of $g$ and $r$ given in the question.

We are given that

$r = 1.6m$

$g = 10m{s^{ - 2}}$

Thus, the minimum speed is ${v_T} = \sqrt {gr} $

$ \Rightarrow {v_T} = \sqrt {10 \times 1.6} $

$ \Rightarrow {v_T} = \sqrt {16} $

$ \Rightarrow {v_T} = 4m{s^{ - 1}}$

Thus, the minimum velocity at the top such that the water from the bucket does not spill should be ${v_T} = 4m{s^{ - 1}}$

**Therefore, option (C) is the correct option.**

**Note:**Do not confuse between uniform and non-uniform circular motion. Ensure that all the quantities are in the same units or SI units before calculations. When a particle or body follows a circular path at constant speed then the body is said to perform uniform circular motion. Though the body moves with constant speed its velocity changes at every point.

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