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.

To find the time taken for ship P to completely pass ship Q when both are moving in the same direction, we can break down the solution step-by-step into three parts: finding the speed of ship P, calculating the speed of ship Q, and determining the relative speed when traveling in the same direction.

Step 1: Calculate the speed of ship P
When a ship crosses a stationary buoy marker, the total distance traveled during this time is simply equal to the length of the ship.
Given:
Length of ship P ($L_P$) = 350 meters
Time taken to cross the buoy marker ($t_1$) = 17.5 seconds

Using the formula for speed:
$\text{Speed of ship P}\left(v_P\right)=\frac{L_P}{t_1}$

Substituting the values:
$v_P=\frac{350}{17.5}=20\text{ m/s}$
Thus, the speed of ship P is 20 meters per second.

Step 2: Calculate the speed of ship Q
When two moving ships completely cross each other, the total distance covered is the sum of their individual lengths.
Total distance ($D$) = Length of ship P + Length of ship Q
$D=350+420=770\text{ meters}$
Time taken to cross each other in opposite directions ($t_2$) = 15.4 seconds

When two objects move in opposite directions, their relative speed is the sum of their individual speeds:
$\text{Relative speed (opposite direction)}=v_P+v_Q=\frac{D}{t_2}$

Substituting the known values:
$20+v_Q=\frac{770}{15.4}$

$20+v_Q=50$

Solving for $v_Q$:
$v_Q=50-20=30\text{ m/s}$
Thus, the speed of ship Q is 30 meters per second.

Step 3: Calculate the time taken when traveling in the same direction
When two objects travel in the same direction, their relative speed is the difference between their individual speeds:
$\text{Relative speed (same direction)}=v_Q-v_P=30-20=10\text{ m/s}$

The total distance required for one ship to completely pass the other remains equal to the sum of their lengths:
$D=770\text{ meters}$

Using the time formula:
$\text{Time taken}=\frac{\text{Total Distance}}{\text{Relative speed (same direction)}}$

$t=\frac{770}{10}=77\text{ seconds}$

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