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**Hint:**For solving the above equation the formula for equation of motion can be used. There are two formulas for the equation of motion. They are non-accelerated motion and the accelerated motion. In case of the accelerated motion the object experiences varied velocity and acceleration. Since the ball with an initial velocity gradually decreases its speed and finally stops, we use the formula of the equation of accelerated motion.

**Useful formula:**

The equation of accelerated motion;

$v = u + at$

Where $v$ denotes the final velocity of the ball, $u$ denotes the initial velocity of the ball, $a$ denotes the acceleration of the ball, $t$ denotes the time taken by the ball.

**Complete step by step solution:**

Given data:

Initial velocity of the ball is $u = 1\,\,m{s^{ - 1}}$ ,

The acceleration of the ball is $a = 0.1\,\,m{s^{ - 2}}$ .

By using the equation for accelerated motion;

$v = u + at$

Substitute the values of initial velocity $u$ and the acceleration $a$ of the ball in the above equation;

Since the ball stops after a certain amount of time the final velocity is $v = 0$.

$0\,\,m{s^{ - 1}} = 1\,\,m{s^{ - 1}} + \left( {0.1\,\,m{s^{ - 2}} \times t} \right)$

On simplifying the equation, we get;

$0 = 1 + 0.1\,t$

$0.1\,t = 1$

Since we only need time taken by the ball;

$t = \dfrac{1}{{0.1}}$

$t = 10\,\,s$

Therefore, the time taken by the ball to stop moving is $t = 10\,\,s$ .

**Hence, the option (C), $t = 10\,\,s$ is the correct answer.**

**Note:**In case of non-accelerated motion, the object experiencing the motion is at constant velocity. There is no change in velocity. The equation of accelerated motion is derived from the basic formula that is acceleration is equal to the change in velocity to the time taken.

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