Given that the bodies A, B, and C have equal kinetic energies, we can use the relationship between kinetic energy ($$K.E.$$) and linear momentum ($$p$$) to find the ratio of their momenta. Recall the formula for kinetic energy is $$K.E. = \frac{1}{2}mv^2$$ and the formula for momentum is $$p = mv$$, where $$m$$ is the mass and $$v$$ is the velocity of the object.

First, from the kinetic energy formula, we can express the velocity in terms of kinetic energy and mass:

$$v = \sqrt{\frac{2 \cdot K.E.}{m}}.$$

The momentum can then be rewritten using the velocity expression obtained from the kinetic energy equation:

$$p = m\sqrt{\frac{2 \cdot K.E.}{m}} = \sqrt{2m \cdot K.E.}.$$

Given that the kinetic energies are the same for all three bodies, we can ignore the kinetic energy term when comparing the ratios, simplifying our comparison to the square root of their masses:

$$p \propto \sqrt{m}.$$

Now, we calculate the ratio of their linear momenta using their masses. Note that the masses should be in consistent units for a valid comparison, so we'll use kilograms for all:

• Mass of A = $$400 \mathrm{~g} = 0.4 \mathrm{~kg}$$


• Mass of B = $$1.2 \mathrm{~kg}$$


• Mass of C = $$1.6 \mathrm{~kg}$$

Thus, the ratio of their momenta will be proportional to the square root of their masses:

$$\text{Ratio of momenta} = \sqrt{0.4} : \sqrt{1.2} : \sqrt{1.6} = \sqrt{\frac{4}{10}} : \sqrt{\frac{12}{10}} : \sqrt{\frac{16}{10}} = \sqrt{\frac{2}{5}} : \sqrt{\frac{6}{5}} : \sqrt{\frac{8}{5}}.$$

Simplifying these we get:

$$\text{Ratio of momenta} = \frac{\sqrt{2}}{\sqrt{5}} : \frac{\sqrt{6}}{\sqrt{5}} : \frac{\sqrt{8}}{\sqrt{5}} = \sqrt{2} : \sqrt{6} : \sqrt{8}.$$

Recognizing that $$\sqrt{6}$$ is equivalent to $$\sqrt{2} \cdot \sqrt{3}$$ and that $$\sqrt{8}$$ is equivalent to $$\sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2\sqrt{2}$$, we see this can also be expressed as:

$$\sqrt{2} : \sqrt{2} \cdot \sqrt{3} : 2\sqrt{2}.$$

Dividing through by $$\sqrt{2}$$ to simplify the ratio, the final ratio of their linear momenta is:

$$1 : \sqrt{3} : 2,$$

which matches Option A $$1: \sqrt{3}: 2.$$