Two trains each of length 100 m are moving with speeds 54 km/h (for Train - 1) and 90 km/h (for Train-2) on two parallel tracks. Match the entries given in column I with the entries given in column II
`{:(" Column I"," Column II"),("(A) Time taken in passing (in s)","(p) 125"),("(B) Distance travelled by "1^(st)" train during passing (in m)","(q) 500"),("(C) Distance travelled by "2^(nd)" train during passing (in m)","(r) 20"),(" if both trains are moving in same direction",),("(D) Displacement of "1^(st)" train w.r.t. 2"^(nd)" train in 2 s (in m)","(s) 300"),(,"(t) 5"):}`

Answer

To solve the problem of matching the entries in Column I with those in Column II, we will follow these steps:

### Step 1: Convert the speeds from km/h to m/s
To convert the speeds of the trains from kilometers per hour to meters per second, we use the conversion factor \( \frac{5}{18} \).

- For Train 1 (speed = 54 km/h):
\[ \text{Speed of Train 1} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

- For Train 2 (speed = 90 km/h):
\[ \text{Speed of Train 2} = 90 \times \frac{5}{18} = 25 \text{ m/s} \]

### Step 2: Calculate the time taken for the trains to pass each other when moving in the same direction
When two objects are moving in the same direction, the relative velocity is the difference of their speeds.

- Relative velocity \( V_r \) when moving in the same direction:
\[ V_r = 25 - 15 = 10 \text{ m/s} \]

- The distance to be covered when passing each other is the sum of their lengths:
\[ \text{Distance} = 100 + 100 = 200 \text{ m} \]

- Time taken to pass each other:
\[ \text{Time} = \frac{\text{Distance}}{V_r} = \frac{200}{10} = 20 \text{ seconds} \]

### Step 3: Calculate the time taken for the trains to pass each other when moving in opposite directions
When moving in opposite directions, the relative velocity is the sum of their speeds.

- Relative velocity \( V_r \) when moving in opposite directions:
\[ V_r = 25 + 15 = 40 \text{ m/s} \]

- Time taken to pass each other:
\[ \text{Time} = \frac{\text{Distance}}{V_r} = \frac{200}{40} = 5 \text{ seconds} \]

### Step 4: Calculate the distance traveled by Train 1 during passing
When the trains are moving in the same direction:
- Time = 20 seconds, Speed of Train 1 = 15 m/s
\[ \text{Distance} = \text{Speed} \times \text{Time} = 15 \times 20 = 300 \text{ m} \]

When the trains are moving in opposite directions:
- Time = 5 seconds, Speed of Train 1 = 15 m/s
\[ \text{Distance} = 15 \times 5 = 75 \text{ m} \]

### Step 5: Calculate the distance traveled by Train 2 during passing
When the trains are moving in the same direction:
- Time = 20 seconds, Speed of Train 2 = 25 m/s
\[ \text{Distance} = 25 \times 20 = 500 \text{ m} \]

When the trains are moving in opposite directions:
- Time = 5 seconds, Speed of Train 2 = 25 m/s
\[ \text{Distance} = 25 \times 5 = 125 \text{ m} \]

### Step 6: Calculate the displacement of Train 1 with respect to Train 2 in 2 seconds
When moving in the same direction:
- Relative velocity \( V_r = 10 \text{ m/s} \)
\[ \text{Displacement} = V_r \times \text{Time} = 10 \times 2 = 20 \text{ m} \]

When moving in opposite directions:
- Relative velocity \( V_r = 40 \text{ m/s} \)
\[ \text{Displacement} = V_r \times \text{Time} = 40 \times 2 = 80 \text{ m} \]

### Final Matching of Columns
Now we can match the results with Column II:
- (A) Time taken in passing (in s) → (p) 20 seconds (same direction) and (t) 5 seconds (opposite direction)
- (B) Distance traveled by 1st train during passing (in m) → (s) 300 m (same direction) and (r) 75 m (opposite direction)
- (C) Distance traveled by 2nd train during passing (in m) → (q) 500 m (same direction) and (p) 125 m (opposite direction)
- (D) Displacement of 1st train w.r.t. 2nd train in 2 s (in m) → (r) 20 m (same direction) and (s) 80 m (opposite direction)