Explanation:
When two masses m1 and m2 collide with each other, the conservation of momentum principle states that the total momentum of the two masses before the collision is equal to the total momentum of the two masses after the collision. If we assume that the masses move in one dimension only, then we can express this principle as follows:

m1u1 + m2u2 = m1v1 + m2v2

Where u1 and u2 are the initial velocities of the masses and v1 and v2 are their final velocities after the collision.

Proportional Ratio:

If we rearrange this equation to solve for the ratio of the change in velocities, we get:

(v1 - u1)/(v2 - u2) = m1/m2

This equation tells us that the ratio of the change in velocities of the two masses is proportional to the ratio of their masses. In other words, if one mass is much larger than the other, then the change in its velocity will be much smaller than the change in the velocity of the smaller mass.

For example, if m1 = 10 kg and m2 = 1 kg, and both masses collide with initial velocities u1 = 2 m/s and u2 = -3 m/s, then we can calculate the final velocities using the conservation of momentum principle:

10 kg * 2 m/s + 1 kg * (-3 m/s) = 10 kg * v1 + 1 kg * v2

20 kg m/s - 3 kg m/s = 10 kg * v1 + 1 kg * v2

17 kg m/s = 10 kg * v1 + 1 kg * v2

If we assume that m1 and m2 stick together after the collision (i.e., they form a single mass), then we can calculate their final velocity using the conservation of energy principle:

(1/2) * (10 kg + 1 kg) * v^2 = (1/2) * 10 kg * 2^2 + (1/2) * 1 kg * (-3)^2

55.5 J = 20 J + 4.5 J

v = sqrt(60/11) m/s

v ≈ 2.32 m/s

Therefore, the change in velocity of m1 is:

v1 - u1 = 2.32 m/s - 2 m/s = 0.32 m/s

And the change in velocity of m2 is:

v2 - u2 = 2.32 m/s - (-3 m/s) = 5.32 m/s

The ratio of these changes is:

0.32 m/s / 5.32 m/s ≈ 0.06

And the ratio of their masses is:

m1 / m2 = 10 kg / 1 kg = 10

Therefore, we can verify that the ratio of the change in velocities is proportional to the ratio of their masses:

0.06 ≈ 10^-1

Conclusion:

In conclusion, when two masses collide with each other, the ratio of the change in their respective velocities is proportional to the ratio of their masses. This principle can be used to predict the final velocities of the masses after the collision.