Transition of electron in an atom and the change in kinetic energy

When an electron undergoes a transition within an atom, it moves from one energy level to another. This transition can result in a change in the electron's kinetic energy. Let's explore the relationship between the initial and final kinetic energies and the change in potential energy.

Initial and final kinetic energy
The initial kinetic energy of the electron (KEi) is given by the equation:

KEi = (1/2)mv₁²

where m is the mass of the electron and v₁ is its initial velocity.

The final kinetic energy of the electron (KEf) is given by the equation:

KEf = (1/2)mv₂²

where v₂ is the final velocity of the electron.

Change in kinetic energy
The change in kinetic energy (∆KE) can be calculated by subtracting the initial kinetic energy from the final kinetic energy:

∆KE = KEf - KEi

Substituting the equations for KEf and KEi, we have:

∆KE = (1/2)mv₂² - (1/2)mv₁²

Simplifying this expression, we get:

∆KE = (1/2)m(v₂² - v₁²)

Since we are given that the final kinetic energy (KEf) is one-fourth of the initial kinetic energy (KEi), we can write:

KEf = (1/4)KEi

Substituting this into the equation for ∆KE, we have:

(1/4)KEi = (1/2)m(v₂² - v₁²)

Multiplying both sides of the equation by 4 to eliminate the fraction, we get:

KEi = 2m(v₂² - v₁²)

Now, let's simplify this equation further to determine the change in kinetic energy.

Simplifying the equation
Expanding the equation, we have:

2m(v₂² - v₁²) = mv₂² - mv₁²

Rearranging the terms, we get:

2m(v₂² - v₁²) - mv₂² = -mv₁²

Simplifying this expression, we have:

mv₂² - 2m(v₁² - v₂²) = -mv₁²

Finally, dividing both sides of the equation by -4m, we get:

(1/4)m(v₁² - v₂²) = (1/4)mv₁²

Conclusion
From the above equation, we can conclude that the change in potential energy (∆PE) is equal to -3/4 times the initial kinetic energy (KEi).

Therefore, the correct answer is option 1) -3/4y.

3/2y