Two particle of masses in the ration 1:2 with charges in the ratio 1:1, are placed at rest in a uniform electric field. They are released and allowed to move for the same time. The ratio of their kinetic energies will be finally

Options

A. `2:1`
B. `8:1`
C. `4:1`
D. `1:4`

Explanation

To solve the problem step by step, we will analyze the motion of the two particles in a uniform electric field and derive the ratio of their kinetic energies.

### Step 1: Define the Variables
Let:
- Mass of particle 1, \( m_1 \)
- Mass of particle 2, \( m_2 = 2m_1 \) (since the ratio of masses is 1:2)
- Charge of particle 1, \( q_1 \)
- Charge of particle 2, \( q_2 = q_1 \) (since the ratio of charges is 1:1)
- Electric field strength, \( E \)

### Step 2: Calculate the Acceleration of Each Particle
The force on each particle due to the electric field is given by:
\[ F = qE \]
Using Newton's second law, \( F = ma \), we can express the acceleration of each particle:
- For particle 1:
\[ a_1 = \frac{q_1 E}{m_1} \]
- For particle 2:
\[ a_2 = \frac{q_2 E}{m_2} = \frac{q_1 E}{2m_1} = \frac{1}{2} \frac{q_1 E}{m_1} \]

### Step 3: Relate the Final Velocity to Acceleration
Since both particles are released from rest and allowed to move for the same time \( t \), we can use the equation of motion:
\[ v = u + at \]
Given that the initial velocity \( u = 0 \):
- For particle 1:
\[ v_1 = a_1 t = \frac{q_1 E}{m_1} t \]
- For particle 2:
\[ v_2 = a_2 t = \frac{1}{2} \frac{q_1 E}{m_1} t \]

### Step 4: Calculate the Linear Momentum of Each Particle
The linear momentum \( p \) of each particle is given by:
\[ p = mv \]
- For particle 1:
\[ p_1 = m_1 v_1 = m_1 \left(\frac{q_1 E}{m_1} t\right) = q_1 E t \]
- For particle 2:
\[ p_2 = m_2 v_2 = 2m_1 \left(\frac{1}{2} \frac{q_1 E}{m_1} t\right) = q_1 E t \]

### Step 5: Find the Ratio of Linear Momenta
Since \( p_1 = p_2 \):
\[ \frac{p_1}{p_2} = \frac{q_1 E t}{q_1 E t} = 1 \]

### Step 6: Calculate the Kinetic Energies
The kinetic energy \( KE \) is given by:
\[ KE = \frac{p^2}{2m} \]
- For particle 1:
\[ KE_1 = \frac{p_1^2}{2m_1} = \frac{(q_1 E t)^2}{2m_1} \]
- For particle 2:
\[ KE_2 = \frac{p_2^2}{2m_2} = \frac{(q_1 E t)^2}{2(2m_1)} = \frac{(q_1 E t)^2}{4m_1} \]

### Step 7: Find the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{(q_1 E t)^2}{2m_1}}{\frac{(q_1 E t)^2}{4m_1}} = \frac{4}{2} = 2 \]

### Final Result
Thus, the ratio of the kinetic energies of particle 1 to particle 2 is:
\[ \frac{KE_1}{KE_2} = 2:1 \]