A train is moving with a speed of 20 m/s at a place where the radius of curvature of the railway line is 1500 m. Calculate the distance between the rails if the elevation of the outer rail above the inner rail is 0.042 m ?

Explanation

To solve the problem, we need to find the distance between the rails (L) given the speed of the train (v), the radius of curvature (R), and the elevation of the outer rail above the inner rail (h).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Speed of the train, \( v = 20 \, \text{m/s} \)
- Radius of curvature, \( R = 1500 \, \text{m} \)
- Elevation of the outer rail above the inner rail, \( h = 0.042 \, \text{m} \)

2. **Use the Relationship Between Elevation and Distance:**
We can use the formula derived from circular motion that relates the angle of elevation (θ), the height (h), and the distance between the rails (L):
\[ \tan^2 \theta = \frac{v^2}{gR} \]
where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

3. **Relate the Tangent to Height and Length:**
From the geometry of the situation, we can express the tangent of the angle as:
\[ \tan \theta = \frac{h}{L} \]

4. **Combine the Two Equations:**
From the two equations, we can set them equal to each other:
\[ \frac{h}{L} = \sqrt{\frac{v^2}{gR}} \]
Rearranging gives:
\[ L = \frac{h \cdot gR}{v^2} \]

5. **Substitute the Known Values:**
Now, we can substitute the known values into the equation:
\[ L = \frac{0.042 \cdot 9.8 \cdot 1500}{20^2} \]

6. **Calculate the Distance:**
First, calculate the numerator:
\[ 0.042 \cdot 9.8 \cdot 1500 = 617.4 \]
Then calculate the denominator:
\[ 20^2 = 400 \]
Now divide:
\[ L = \frac{617.4}{400} = 1.5435 \, \text{m} \]

7. **Round the Result:**
Rounding to two decimal places, we find:
\[ L \approx 1.54 \, \text{m} \]

### Final Answer:
The distance between the rails is approximately **1.54 meters**.