Explanation:

To find the ratio of velocities of the proton, deuteron, and alpha particle, we need to consider the relationship between potential energy and kinetic energy.

The potential energy of a charged particle accelerated through a potential difference is given by the equation:

PE = qV

Where PE is the potential energy, q is the charge of the particle, and V is the potential difference.

The kinetic energy of a particle is given by the equation:

KE = (1/2)mv^2

Where KE is the kinetic energy, m is the mass of the particle, and v is its velocity.

From the above equations, we can see that the potential energy is directly proportional to the charge and potential difference, while the kinetic energy is directly proportional to the mass and the square of the velocity.

Ratio of Velocities:

Let's assume the ratio of velocities of the proton, deuteron, and alpha particle to be v₁, v₂, and v₃ respectively.

Given:
Potential difference for proton (V₁) = 1V
Potential difference for deuteron (V₂) = 2V
Potential difference for alpha particle (V₃) = 4V

We can write the equations for potential energy as follows:

PE₁ = q₁V₁
PE₂ = q₂V₂
PE₃ = q₃V₃

Since the potential energy is the same for all the particles, we can equate them:

q₁V₁ = q₂V₂ = q₃V₃

Now, let's consider the kinetic energy equations:

KE₁ = (1/2)m₁v₁²
KE₂ = (1/2)m₂v₂²
KE₃ = (1/2)m₃v₃²

Since the potential energy is converted into kinetic energy, we can equate the potential energy equations with the kinetic energy equations:

q₁V₁ = (1/2)m₁v₁²
q₂V₂ = (1/2)m₂v₂²
q₃V₃ = (1/2)m₃v₃²

Calculating the Ratios:

Now, we can calculate the ratio of velocities by dividing the kinetic energy equations:

v₁²/v₂² = (q₁V₁)/(q₂V₂)
v₂²/v₃² = (q₂V₂)/(q₃V₃)

Taking the square root of both sides, we get:

v₁/v₂ = √(q₁V₁)/(q₂V₂)
v₂/v₃ = √(q₂V₂)/(q₃V₃)

Substituting the given values for potential difference (V₁, V₂, V₃) and charges (q₁, q₂, q₃), we can calculate the ratio of velocities.