Two trains `A` and `B` `, 100km` apart are travelling towards each other on different tracks with same starting speed of `50km//h`. The train `A` accelerates at `20km//h^(2)` and the train `B` retards at the rate `20km//h^(2)`. The distance covered by the train `A` when they cross each other is

Options

A. `70km`
B. `55km`
C. `65km`
D. `60km`

Answer

To solve the problem of two trains A and B traveling towards each other, we can break down the solution into a series of steps.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Train A starts with an initial speed \( u_A = 50 \, \text{km/h} \) and accelerates at \( a_A = 20 \, \text{km/h}^2 \).
- Train B starts with an initial speed \( u_B = 50 \, \text{km/h} \) and decelerates at \( a_B = -20 \, \text{km/h}^2 \).
- The distance between the two trains is \( D = 100 \, \text{km} \).

2. **Set Up the Distance Equations**:
- The distance covered by Train A when they cross each other can be denoted as \( h \).
- The distance covered by Train B will then be \( 100 - h \).

3. **Use the Equations of Motion**:
- For Train A, the distance covered can be expressed as:
\[ h = u_A t + \frac{1}{2} a_A t^2 \]
- For Train B, the distance covered can be expressed as:
\[ 100 - h = u_B t + \frac{1}{2} a_B t^2 \]

4. **Substitute the Known Values**:
- For Train A:
\[ h = 50t + \frac{1}{2} (20)t^2 \quad \text{(Equation 1)} \]
- For Train B:
\[ 100 - h = 50t - \frac{1}{2} (20)t^2 \quad \text{(Equation 2)} \]

5. **Combine the Equations**:
- Adding Equation 1 and Equation 2:
\[ h + (100 - h) = 50t + \frac{1}{2}(20)t^2 + 50t - \frac{1}{2}(20)t^2 \]
- This simplifies to:
\[ 100 = 100t \]
- Therefore, we find:
\[ t = 1 \, \text{hour} \]

6. **Calculate the Distance Covered by Train A**:
- Substitute \( t = 1 \) back into Equation 1:
\[ h = 50(1) + \frac{1}{2}(20)(1^2) \]
- This gives:
\[ h = 50 + 10 = 60 \, \text{km} \]

7. **Conclusion**:
- The distance covered by Train A when they cross each other is \( 60 \, \text{km} \).

### Final Answer:
The distance covered by Train A when they cross each other is **60 km**.

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