Solution:

Given, the average rotational kinetic energy of hydrogen molecule at a temperature T is E.

The average translational kinetic energy of helium at the same temperature will be found.

We know that the total kinetic energy of a gas is the sum of its translational, rotational and vibrational kinetic energies.

Total kinetic energy of hydrogen molecule,

K = translational KE + rotational KE + vibrational KE

Since hydrogen molecule is diatomic, it has three degrees of freedom.

Hence, the translational kinetic energy of hydrogen molecule is given by

Ktrans = (3/2)kT

Rotational kinetic energy of hydrogen molecule is given by

Krot = (1/2)Iω^2

where I is the moment of inertia and ω is the angular velocity.

For a hydrogen molecule, moment of inertia is given by

I = 2mr^2

where m is the mass of hydrogen molecule and r is the bond length.

Therefore, the rotational kinetic energy of hydrogen molecule is given by

Krot = (1/2)(2mr^2)(ω^2)

Krot = (1/2)(2mr^2)((2kT)/I)

Krot = (kT)

Total kinetic energy of hydrogen molecule is given by

K = (3/2)kT + (1/2)kT + vibrational KE

K = (5/2)kT + vibrational KE

Given, the average rotational kinetic energy of hydrogen molecule at a temperature T is E.

Hence, Krot = E

Therefore, K = (3/2)kT + E + vibrational KE

Now, let's find the average translational kinetic energy of helium at the same temperature T.

Helium is a monatomic gas and has three degrees of freedom.

Hence, the translational kinetic energy of helium is given by

Ktrans = (3/2)kT

Therefore, the total kinetic energy of helium is given by

K = (3/2)kT + vibrational KE

Since both hydrogen and helium are at the same temperature T, their vibrational kinetic energies will be the same.

Hence, we can equate their total kinetic energies to find the ratio of their translational kinetic energies.

(5/2)kT + E + vibrational KE = (3/2)kT + vibrational KE

(5/2)kT + E = (3/2)kT

E = (3/2)kT

Therefore, the average translational kinetic energy of helium at the same temperature will be 3E/2.

Hence, option D is the correct answer.