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**Hint:**The question is asking about the impulse, to find the value of impulse, we first need to know that impulse is nothing but the difference in the initial and final momentum of the object. Hence, to find the impulse imparted, you need to find the initial momentum and the final momentum of both the balls and then find the difference between them to find the value of impulse.

**Complete step by step answer:**

We will approach the solution to question exactly as explained in the hint section of the solution. To find the impulse imparted on each ball due to the other, we simply need to find the initial and final momentum of both the balls and find the difference between them to find the value of impulse imparted on each ball due to the other.

Momentum can be given as:

$P = mv$

Where $P$ is the momentum of the object,

$m$ is the mass of the object and,

$v$ is the velocity of the object

Let us name the two billiards balls as ball $1$ and ball $2$

The question has told us that mass of both the balls is same and given as:

${m_1} = {m_2} = 0.06\,kg$

Similarly, both of the balls have the exact same initial velocities given to be:

$

{u_1} = 5\,m{s^{ - 1}} \\

{u_2} = - 5\,m{s^{ - 1}} \\

$

Note that initial velocity of the second ball is negative since both the balls are moving in the opposite directions

Now, if we approach the final velocities of each balls, it has been told to us that the magnitude stays the same while the direction becomes completely opposite, hence, we can write:

$

{v_1} = - 5\,m{s^{ - 1}} \\

{v_2} = 5\,m{s^{ - 1}} \\

$

Now, using the given information, we can find the initial momentums of both the balls as:

$

{P_{1i}} = {m_1}{u_1} \\

{P_{2i}} = {m_2}{u_2} \\

$

Substituting the values, we get:

$

{P_{1i}} = \left( {0.06} \right)\left( 5 \right) \\

{P_{2i}} = \left( {0.06} \right)\left( { - 5} \right) \\

$

Solving, we get:

$

{P_{1i}} = 0.3 \\

{P_{2i}} = - 0.3 \\

$

Now, let us find the final momentums of both the balls as:

$

{P_{1f}} = {m_1}{v_1} \\

{P_{2f}} = {m_2}{v_2} \\

$

Upon substituting the values, we get:

$

{P_{1f}} = \left( {0.06} \right)\left( { - 5} \right) \\

{P_{2f}} = \left( {0.06} \right)\left( 5 \right) \\

$

Upon solving, we get:

$

{P_{1f}} = - 0.3 \\

{P_{2f}} = 0.3 \\

$

Now, if we find the impulse on both the balls, we get:

$

{I_1} = {P_{1f}} - {P_{1i}} \\

{I_2} = {P_{2f}} - {P_{2i}} \\

$

If we substitute the values of the momentums that we found out, we get:

$

{I_1} = \left( { - 0.3} \right) - 0.3 \\

{I_2} = 0.3 - \left( { - 0.3} \right) \\

$

Upon solving, we get:

$

{I_1} = - 0.6 \\

{I_2} = 0.6 \\

$

Hence, we can see that the impulse imparted on both the balls due to each other is same in the magnitude but exactly opposite in the sense of direction

**Hence, the correct answer is option (B), i.e. $0.6{\kern 1pt} kg{\kern 1pt} m{s^{ - 1}}$ each, in opposite directions.**

**Note:**Major mistake that most of the students make is that they do not consider the direction of the motion of each of the balls and treat their velocities as scalar instead of vector. If you make the same mistake, you would get zero and the answer, which is completely wrong and should never be followed.

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