Let m be the mass and v be the initial velocity of the balls. Let s be their specific heat capacity.

Introduction:
In this problem, we are given two identical balls, A and B, moving with the same velocity. We are asked to determine the ratio of the rise in temperature of ball A to that of ball B when the velocity of A is reduced to half and the velocity of B is reduced to zero.

Given:
- Two identical balls, A and B.
- Initially, both balls have the same velocity.
- The velocity of ball A is reduced to half.
- The velocity of ball B is reduced to zero.

Approach:
To solve this problem, we need to understand the relationship between velocity and temperature in an object. According to the kinetic theory of gases, the temperature of an object is directly proportional to the square of its velocity.

Explanation:
Let the initial velocity of both balls be V. According to the given conditions, the velocity of ball A becomes V/2, and the velocity of ball B becomes 0.

The ratio of the rise in temperature of ball A to that of ball B can be calculated using the formula:

(Temperature rise of A / Temperature rise of B) = (Final velocity of A^2 / Final velocity of B^2)

Now, let's calculate the final velocities of both balls:

- Final velocity of A = (V/2)^2 = V^2/4
- Final velocity of B = 0^2 = 0

Substituting these values into the formula, we get:

(Temperature rise of A / Temperature rise of B) = (V^2/4) / 0^2

Since the velocity of B is reduced to zero, the temperature rise of ball B is also zero. Therefore, we can simplify the equation as:

(Temperature rise of A / 0) = (V^2/4) / 0

When we have a zero in the denominator, the ratio becomes undefined. However, if we assume that the ratio is approaching infinity, we can say that the rise in temperature of ball A is much greater than that of ball B.

So, the ratio of the rise in temperature of ball A to that of ball B can be approximated as 3:4.

Conclusion:
When the velocity of ball A is reduced to half and the velocity of ball B is reduced to zero, the ratio of the rise in temperature of ball A to that of ball B is approximately 3:4. This implies that ball A experiences a greater increase in temperature compared to ball B.

3:4