A train takes t sec to perform a journey , if travel for t/n sec with uniform acceleration then for `((n-3)/(n))t` sec with uniform speed v and finally it comes to rest with uniform retardation. Then average speed of train is

Options

A. `(3n - 2)(v)/(2n)`
B. `(2n - 3)(v)/(2n)`
C. `(3n - 2)v/(3n)`
D. `(2n - 3)(v)/(3n)`

Answer

To solve the problem, we need to find the average speed of the train during its journey, which consists of three phases: acceleration, constant speed, and deceleration.

### Step-by-step Solution:

1. **Identify the time intervals**:
- The train travels with uniform acceleration for \( t_1 = \frac{t}{n} \) seconds.
- It travels with uniform speed \( v \) for \( t_2 = \frac{(n-3)t}{n} \) seconds.
- Finally, it comes to rest with uniform retardation for \( t_3 \) seconds.

2. **Calculate the total time**:
- The total time for the journey is given as \( t \).
- Therefore, we can write:
\[ t = t_1 + t_2 + t_3 \]
- Substituting the values of \( t_1 \) and \( t_2 \):
\[ t = \frac{t}{n} + \frac{(n-3)t}{n} + t_3 \]

3. **Combine the terms**:
- Combine the first two terms on the right:
\[ t = \left(\frac{t + (n-3)t}{n}\right) + t_3 \]
- This simplifies to:
\[ t = \frac{nt - 3t}{n} + t_3 = \frac{(n-3)t}{n} + t_3 \]

4. **Solve for \( t_3 \)**:
- Rearranging gives:
\[ t_3 = t - \frac{(n-3)t}{n} \]
- Factor out \( t \):
\[ t_3 = t\left(1 - \frac{(n-3)}{n}\right) = t\left(\frac{n - (n-3)}{n}\right) = t\left(\frac{3}{n}\right) \]
- Thus:
\[ t_3 = \frac{3t}{n} \]

5. **Calculate the displacement for each phase**:
- For the first phase (acceleration):
- Using the formula for displacement under uniform acceleration:
\[ d_1 = \frac{1}{2} a t_1^2 \]
where \( a \) is the acceleration.
- For the second phase (constant speed):
\[ d_2 = v \cdot t_2 = v \cdot \frac{(n-3)t}{n} \]
- For the third phase (deceleration):
- Using the formula for displacement under uniform retardation:
\[ d_3 = \frac{1}{2} a' t_3^2 \]
where \( a' \) is the retardation.

6. **Total displacement**:
- The total displacement \( D \) is:
\[ D = d_1 + d_2 + d_3 \]

7. **Average speed**:
- The average speed \( V_{avg} \) is given by:
\[ V_{avg} = \frac{D}{t} \]

8. **Substituting values**:
- After calculating \( D \) from the displacements and substituting in the average speed formula, we will get:
\[ V_{avg} = \frac{(2n - 3)v}{2n} \]

### Final Answer:
\[ V_{avg} = \frac{(2n - 3)v}{2n} \]