A train starting from rest, and moving with uniform acceleration `alpha`, acquires a speed v. Then its comes to a stop with uniform retardation `beta`. Then the average velocity of the train is

Options

A. `v//2`
B. `v//4`
C. `(alpha + beta)t//2`
D. `(alpha + beta)t//alphabeta`

Explanation

To find the average velocity of the train, we will follow these steps:

### Step 1: Calculate the Displacement during Acceleration
The train starts from rest and accelerates uniformly with acceleration \( \alpha \) until it reaches a speed \( v \). We can use the kinematic equation:

\[ v^2 = u^2 + 2a s \]

Where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity (which is 0 since the train starts from rest),
- \( a \) is the acceleration (\( \alpha \)),
- \( s \) is the displacement during acceleration (\( s_1 \)).

Substituting the values:

\[ v^2 = 0 + 2 \alpha s_1 \implies s_1 = \frac{v^2}{2\alpha} \]

### Step 2: Calculate the Displacement during Retardation
Next, the train comes to a stop with uniform retardation \( \beta \). We can again use the kinematic equation:

\[ v^2 = u^2 + 2a s \]

This time, the initial velocity \( u \) is \( v \), the final velocity is 0, and the acceleration is \( -\beta \). Thus, we have:

\[ 0 = v^2 - 2\beta s_2 \implies s_2 = \frac{v^2}{2\beta} \]

### Step 3: Calculate Total Displacement
The total displacement \( S \) of the train is the sum of the displacements during acceleration and retardation:

\[ S = s_1 + s_2 = \frac{v^2}{2\alpha} + \frac{v^2}{2\beta} \]

Factoring out \( \frac{v^2}{2} \):

\[ S = \frac{v^2}{2} \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \]

### Step 4: Calculate Time Taken during Acceleration
The time taken during acceleration \( t_1 \) can be calculated using:

\[ v = u + at \implies t_1 = \frac{v - 0}{\alpha} = \frac{v}{\alpha} \]

### Step 5: Calculate Time Taken during Retardation
The time taken during retardation \( t_2 \) can be calculated similarly:

\[ 0 = v - \beta t \implies t_2 = \frac{v}{\beta} \]

### Step 6: Calculate Total Time Taken
The total time \( T \) taken by the train is the sum of the times during acceleration and retardation:

\[ T = t_1 + t_2 = \frac{v}{\alpha} + \frac{v}{\beta} \]

Factoring out \( v \):

\[ T = v \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \]

### Step 7: Calculate Average Velocity
The average velocity \( V_{avg} \) is given by the formula:

\[ V_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{S}{T} \]

Substituting the values of \( S \) and \( T \):

\[ V_{avg} = \frac{\frac{v^2}{2} \left( \frac{1}{\alpha} + \frac{1}{\beta} \right)}{v \left( \frac{1}{\alpha} + \frac{1}{\beta} \right)} \]

Simplifying this expression:

\[ V_{avg} = \frac{v^2}{2v} = \frac{v}{2} \]

### Final Answer
Thus, the average velocity of the train is:

\[ \boxed{\frac{v}{2}} \]