if `m_p` and `m_(alpha)` are masses of proton and `alpha-`particles find ratio of momentum of a proton and an `alpha-`particle which are accelerated from rest by a potential difference of 200 V.

Options

A. `(m_(p))/(2m_(alpha))`
B. `(2m_(alpha))/(m_(p))`
C. `(sqrt((m_(p))/(2m_(alpha))))`
D. `sqrt((2m_(p))/(m_(alpha)))`

Explanation

To find the ratio of momentum of a proton and an alpha particle when both are accelerated from rest by a potential difference of 200 V, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and momentum
The kinetic energy (KE) of a particle can be expressed in terms of its momentum (p) as:
\[ KE = \frac{p^2}{2m} \]
where \( m \) is the mass of the particle.

### Step 2: Relate kinetic energy to the potential difference
When a charged particle is accelerated through a potential difference \( V \), its kinetic energy can also be expressed as:
\[ KE = qV \]
where \( q \) is the charge of the particle.

### Step 3: Equate the two expressions for kinetic energy
From the two expressions for kinetic energy, we have:
\[ \frac{p^2}{2m} = qV \]

### Step 4: Solve for momentum
Rearranging this equation gives us:
\[ p^2 = 2mqV \]
Taking the square root of both sides, we find:
\[ p = \sqrt{2mqV} \]

### Step 5: Calculate momentum for proton and alpha particle
For a proton:
- Mass = \( m_p \)
- Charge = \( q_p \) (the charge of a proton)

Thus, the momentum of the proton is:
\[ p_p = \sqrt{2m_p q_p V} \]

For an alpha particle:
- Mass = \( m_\alpha \)
- Charge = \( q_\alpha = 2q_p \) (the charge of an alpha particle is twice that of a proton)

Thus, the momentum of the alpha particle is:
\[ p_\alpha = \sqrt{2m_\alpha q_\alpha V} = \sqrt{2m_\alpha (2q_p) V} = \sqrt{4m_\alpha q_p V} \]

### Step 6: Find the ratio of momenta
Now, we can find the ratio of the momentum of the proton to that of the alpha particle:
\[ \frac{p_p}{p_\alpha} = \frac{\sqrt{2m_p q_p V}}{\sqrt{4m_\alpha q_p V}} \]

### Step 7: Simplify the ratio
The \( q_p V \) terms cancel out:
\[ \frac{p_p}{p_\alpha} = \frac{\sqrt{2m_p}}{\sqrt{4m_\alpha}} = \frac{\sqrt{2m_p}}{2\sqrt{m_\alpha}} = \frac{\sqrt{m_p}}{\sqrt{2m_\alpha}} \]

### Final Result
Thus, the ratio of the momentum of a proton to that of an alpha particle is:
\[ \frac{p_p}{p_\alpha} = \frac{\sqrt{m_p}}{\sqrt{2m_\alpha}} \]