(a) v/c

for kinetic energy E, Momentum P, Mp is mass of proton and Me is mass of electron ,Vp is speed of proton
for same De Broglie wavelength Lambda,

lambda =h/√(2mE)
So Ee×Me=Ep×Mp--------------(1)

also Lambda =h/mv
So
Me/Mp= Vp/Ve
Me/Mp=c/v ----------------------(2)

Solving (1)&(2) we get,
Ee/Ep=v/c

De Broglie Wavelength and Energy

The De Broglie wavelength, denoted by λ, is a concept in quantum mechanics that relates the momentum and wavelength of a particle. It is given by the equation:

λ = h / p

where λ is the De Broglie wavelength, h is the Planck's constant, and p is the momentum of the particle.

The momentum of a particle can be calculated as:

p = mv

where m is the mass of the particle and v is its velocity.

Equal De Broglie Wavelengths

Given that the electron and proton possess the same De Broglie wavelength, we can equate their momentum equations:

melectron * velectron = mproton * vproton

where melectron and mproton are the masses of the electron and proton respectively, and velectron and vproton are their velocities.

Comparing Energies

The energy of a particle can be calculated using the equation:

E = mc^2

where E is the energy, m is the mass of the particle, and c is the speed of light.

For the electron, we have:

Ee = melectron * c^2

For the proton, we have:

Ep = mproton * c^2

To compare the energies, we can calculate the ratio:

Ee / Ep = (melectron * c^2) / (mproton * c^2)

Simplifying the equation, we find:

Ee / Ep = melectron / mproton

Ratio of Electron and Proton Energies

Since the electron and proton possess the same De Broglie wavelength, their momentum equation can be rearranged as:

velectron / vproton = mproton / melectron

Substituting this into the ratio of energies equation, we get:

Ee / Ep = melectron / mproton = velectron / vproton

Therefore, the ratio of the electron energy to the proton energy is equal to the ratio of their velocities:

Ee / Ep = velectron / vproton

Answer

Therefore, the correct answer is (a) v / c. The ratio of the electron energy to the proton energy is equal to the ratio of their velocities, which is v / c.