Introduction:
In this scenario, we have two balls, A and B, with masses m and 2m, respectively. Both balls are in motion with velocities 2v and v, respectively. We need to compare the force required to stop both balls in the same time.

Explanation:
To compare the force required to stop the balls, we need to understand the concept of Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration.

Force on Ball A:
Using Newton's second law, we can calculate the force required to stop ball A. The acceleration of ball A can be determined by the change in velocity over time. Since the ball is brought to rest, its final velocity is 0. Therefore, the acceleration of ball A is given by:

acceleration (a) = (final velocity - initial velocity) / time
= (0 - 2v) / t
= -2v / t

The force required to stop ball A can be calculated using Newton's second law:

force on ball A = mass of ball A * acceleration
= m * (-2v / t)
= -2mv / t

Force on Ball B:
Using the same approach, we can calculate the force required to stop ball B. The acceleration of ball B is given by:

acceleration (a) = (final velocity - initial velocity) / time
= (0 - v) / t
= -v / t

The force required to stop ball B can be calculated using Newton's second law:

force on ball B = mass of ball B * acceleration
= 2m * (-v / t)
= -2mv / t

Comparing the Forces:
Now, let's compare the forces required to stop both balls:

force on ball A : force on ball B
= (-2mv / t) : (-2mv / t)
= 1 : 1

Therefore, the force required to stop both balls A and B in the same time is in the ratio of 1:1.

Conclusion:
The correct answer is option 'c' - 1:1. The force required to stop balls A and B in the same time is equal.