A cargo ship P, which is 350 meters in length, crosses a stationary buoy marker in 17.5 seconds. It then crosses another cargo ship Q, which is 420 meters long and moving in the direction opposite to P, in 15.4 seconds. In how much time will ship P completely pass ship Q if both ships are traveling in the same direction?

The correct answer is 77 seconds.

Step 1: Calculate the speed of cargo ship P
When ship P crosses a stationary buoy marker, the total distance covered is equal to the length of ship P itself.
Length of ship P = 350 meters
Time taken to cross the buoy marker = 17.5 seconds

$Speed\; of\; ship\; P=\frac{350}{17.5}=20\text{m/s}$

Step 2: Calculate the speed of cargo ship Q
When ship P crosses ship Q moving in the opposite direction, the relative speed is the sum of their speeds:
Relative Speed (opposite) = Speed of P + Speed of Q
Total distance to cross each other = Length of P + Length of Q = 350 + 420 = 770 meters
Time taken = 15.4 seconds

$\text{Relative Speed (opposite)}=\frac{770}{15.4}=50\text{m/s}$

Using the speed of ship P (20 m/s):

$20+\text{Speed of Q}=50\Rightarrow \text{Speed of Q}=50-20=30\text{m/s}$

Step 3: Calculate the time needed for ship P to pass ship Q in the same direction
When both ships move in the same direction, the relative speed is the difference between their speeds:
Relative Speed (same direction) = Speed of Q - Speed of P = 30 - 20 = 10 m/s
Total distance to completely pass each other remains the combined length of both ships = 770 meters.

$\text{Time required}=\frac{\text{Total Distance}}{\text{Relative Speed (same direction)}}=\frac{770}{10}=77\text{seconds}$

Thus, ship P will completely pass ship Q in 77 seconds.