An electron of mass `m_e` starts from rest, moves a certain distance in uniform field and takes time `t_1`. A proton of mass `m_p` also starts from rest in the same field and covers the same distande in time `t_2` ,then `t_1//t_2` is equal to

Options

A. `sqrt(m_e/m_p)`
B. `sqrt(m_p/m_e)`
C. 1
D. 1836

Explanation

To solve the problem, we need to analyze the motion of both the electron and the proton in a uniform electric field. Here’s a step-by-step solution:

### Step 1: Understand the Forces Acting on the Particles
Both the electron and the proton experience a force due to the electric field \( E \). The force \( F \) acting on a charged particle in an electric field is given by:
\[ F = qE \]
where \( q \) is the charge of the particle.

### Step 2: Determine the Acceleration of Each Particle
The acceleration \( a \) of each particle can be found using Newton's second law:
\[ F = ma \]
Thus, we can express the acceleration as:
\[ a = \frac{F}{m} = \frac{qE}{m} \]
For the electron (charge \( -e \), mass \( m_e \)):
\[ a_e = \frac{eE}{m_e} \]
For the proton (charge \( +e \), mass \( m_p \)):
\[ a_p = \frac{eE}{m_p} \]

### Step 3: Use Kinematic Equations
Since both particles start from rest and travel the same distance \( s \), we can use the kinematic equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Given that the initial velocity \( u = 0 \), this simplifies to:
\[ s = \frac{1}{2} a t^2 \]
For the electron:
\[ s = \frac{1}{2} a_e t_1^2 = \frac{1}{2} \left(\frac{eE}{m_e}\right) t_1^2 \]
For the proton:
\[ s = \frac{1}{2} a_p t_2^2 = \frac{1}{2} \left(\frac{eE}{m_p}\right) t_2^2 \]

### Step 4: Set the Distances Equal
Since both particles travel the same distance \( s \), we can set the two equations equal to each other:
\[ \frac{1}{2} \left(\frac{eE}{m_e}\right) t_1^2 = \frac{1}{2} \left(\frac{eE}{m_p}\right) t_2^2 \]
The \( \frac{1}{2} \) and \( eE \) terms cancel out:
\[ \frac{t_1^2}{m_e} = \frac{t_2^2}{m_p} \]

### Step 5: Solve for the Ratio of Times
Rearranging gives:
\[ \frac{t_1^2}{t_2^2} = \frac{m_e}{m_p} \]
Taking the square root of both sides:
\[ \frac{t_1}{t_2} = \sqrt{\frac{m_e}{m_p}} \]

### Final Answer
Thus, the ratio of the times \( t_1 \) and \( t_2 \) is:
\[ \frac{t_1}{t_2} = \sqrt{\frac{m_e}{m_p}} \]