1). Whatever kinetic energy the rolling ball has is the amount of energy you have to absorb in order to stop it.
2). Whatever momentum the rolling ball has is the amount of momentum you have to provide in the other direction to cancel it.
Since you asked about force and time, we sense 'impulse' in the air, and we know that impulse is exactly a change in momentum. So let's use #2 and talk about momentum and impulse.
Impulse = (force) x (time)
Momentum of a moving object is (mass) x (speed) .
-- Momentum of the first ball: (8 kg) x (0.2 m/s) = 1.6 kg-m/s
Impulse required to stop it = 1.6 kg-m/s
(force) x (10 sec) = 1.6
Force required = 1.6 / 10 = 0.16 Newton .
-- Momentum of the second ball: (4 kg) x (1 m/s) = 4 kg-m/s
Impulse required to stop it = 4 kg-m/s
(force) x (10 sec) = 4
Force required = 4 / 10 = 0.4 Newton .
You need more force o stop the second ball. Although its mass is only 1/2 the mass of the 8kg ball, it's moving 5 times as fast, and has 2.5 times the momentum of the bigger ball. So you need 2.5 times as much impulse to stop it. If you're going to push on each ball for the same length of time, then you need to push 2.5 times as hard on the smaller ball in order to stop it.
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