Introduction:
Momentum and kinetic energy are two fundamental concepts in physics. While momentum is the product of an object's mass and velocity, kinetic energy is the energy possessed by an object due to its motion.

Explanation:
When comparing the kinetic energies of a light and a heavy object with the same momentum, it is essential to understand the relationship between mass, velocity, momentum, and kinetic energy.

1. The relationship between momentum and kinetic energy:
Momentum (p) is given by the equation p = m * v, where m represents the mass of the object and v represents its velocity. Kinetic energy (KE) is expressed as KE = 0.5 * m * v^2. From these equations, we can observe that both momentum and kinetic energy depend on mass and velocity.

2. The ratio of kinetic energies:
Let's consider two objects, one light and one heavy, with the same momentum. Since momentum is equal, we can equate the two expressions: m1 * v1 = m2 * v2. Rearranging this equation, we find that v2/v1 = m1/m2.

To calculate the ratio of their kinetic energies, we substitute the values of mass and velocity into the kinetic energy equation. The ratio of their kinetic energies (KE1/KE2) can be calculated as (0.5 * m1 * v1^2) / (0.5 * m2 * v2^2). Simplifying this expression, we find that KE1/KE2 = v1^2/v2^2 = (v1/v2)^2 = (m2/m1)^2.

3. Comparison of kinetic energies:
Since the ratio of kinetic energies is proportional to the square of the ratio of masses, we can conclude that the heavier object has a larger kinetic energy. This is because the kinetic energy is directly proportional to mass, and the square term magnifies the effect.

Conclusion:
In conclusion, when two objects with the same momentum are compared, the object with a higher mass will have a larger kinetic energy. This is due to the direct proportionality between mass and kinetic energy, as well as the effect of the square term in the ratio of kinetic energies.