An elevator of height `'h'` ascends with constant acceleration `'a'`. When it corsses a plalform bolt drops from the top of the elevator. If the time for the bolt to hit the floor of the elevator is `sqrt((lambdah)/(g + a))` then find `'lambda'`.

Explanation

To solve the problem, we need to analyze the motion of the bolt dropped from the elevator. The elevator is ascending with a constant acceleration 'a', and we need to find the time it takes for the bolt to hit the floor of the elevator.

### Step-by-Step Solution:

1. **Identify the Situation**:
- The elevator has a height 'h' and is moving upwards with an acceleration 'a'.
- The bolt is dropped from the top of the elevator when it crosses a platform.

2. **Determine the Forces Acting on the Bolt**:
- Once the bolt is dropped, it is only influenced by gravity. The acceleration due to gravity is 'g' acting downwards.
- The elevator continues to accelerate upwards with 'a'.

3. **Calculate the Relative Acceleration**:
- The relative acceleration of the bolt with respect to the elevator can be calculated as:
\[ a_{\text{relative}} = g + a \]
- Here, 'g' is positive (downward) and 'a' is also considered positive (upward), hence the total effective acceleration acting on the bolt relative to the elevator is \( g + a \).

4. **Use the Equation of Motion**:
- The distance the bolt falls relative to the elevator is 'h'. Using the second equation of motion:
\[ h = \frac{1}{2} a_{\text{relative}} t^2 \]
- Substituting the relative acceleration:
\[ h = \frac{1}{2} (g + a) t^2 \]

5. **Rearranging the Equation**:
- Rearranging the equation to solve for time 't':
\[ t^2 = \frac{2h}{g + a} \]
- Taking the square root gives:
\[ t = \sqrt{\frac{2h}{g + a}} \]

6. **Comparing with Given Time Expression**:
- The problem states that the time 't' is also given by:
\[ t = \sqrt{\frac{\lambda h}{g + a}} \]
- Setting the two expressions for 't' equal to each other:
\[ \sqrt{\frac{2h}{g + a}} = \sqrt{\frac{\lambda h}{g + a}} \]

7. **Squaring Both Sides**:
- Squaring both sides to eliminate the square root:
\[ \frac{2h}{g + a} = \frac{\lambda h}{g + a} \]

8. **Canceling Common Terms**:
- Since \( h \) and \( g + a \) are positive and non-zero, we can cancel them:
\[ 2 = \lambda \]

9. **Final Result**:
- Therefore, the value of \( \lambda \) is:
\[ \lambda = 2 \]

### Conclusion:
The value of \( \lambda \) is 2.

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