The two ends of a train moving with constant acceleration pass a certain point with velocities u and 3u. The velocity with which the middle point of the train passes the same point is

Options

A. 2u
B. `(3)/(2)u`
C. `sqrt5u`
D. `sqrt(10)u`

Answer

To solve the problem, we need to determine the velocity of the middle point of the train as it passes a certain point, given that the two ends of the train pass that point with velocities \( u \) and \( 3u \) respectively.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The front end of the train passes the observation point with velocity \( u \).
- The rear end of the train passes the same point with velocity \( 3u \).
- The train is moving with constant acceleration.

2. **Identifying Variables**:
- Let the length of the train be \( l \).
- The initial velocity of the front end (engine) is \( u \).
- The final velocity of the rear end is \( 3u \).

3. **Using the Kinematic Equation**:
- We can use the equation of motion:
\[ v^2 = u^2 + 2as \]
- Here, \( v \) is the final velocity (which is \( 3u \)), \( u \) is the initial velocity (which is \( u \)), \( a \) is the acceleration, and \( s \) is the distance covered (which is the length of the train, \( l \)).
- Plugging in the values:
\[ (3u)^2 = u^2 + 2a(l) \]
\[ 9u^2 = u^2 + 2al \]
- Rearranging gives:
\[ 8u^2 = 2al \implies a = \frac{4u^2}{l} \]

4. **Finding the Velocity of the Middle Point**:
- The middle point of the train will pass the observation point when the front end has covered half the length of the train, i.e., \( \frac{l}{2} \).
- We need to find the velocity of the front end when it has covered this distance.
- Using the same kinematic equation:
\[ v^2 = u^2 + 2as \]
- Here, \( s = \frac{l}{2} \):
\[ v^2 = u^2 + 2\left(\frac{4u^2}{l}\right)\left(\frac{l}{2}\right) \]
\[ v^2 = u^2 + 4u^2 = 5u^2 \]
- Taking the square root gives:
\[ v = \sqrt{5}u \]

5. **Conclusion**:
- The velocity with which the middle point of the train passes the observation point is \( \sqrt{5}u \).

### Final Answer:
The velocity with which the middle point of the train passes the same point is \( \sqrt{5}u \).

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