Given information:
- Ship A is moving westward with a speed of 10 km/hr.
- Ship B is 100 km south of ship A and is moving northward with a speed of 10 km/hr.

Objective:
To find the time at which the distance between the two ships becomes shortest.

Approach:
We can solve this problem by considering the relative motion of the two ships. We will track the positions of the two ships at different times and calculate the distance between them.

Calculating the positions of the ships:
Since ship A is moving westward, its position can be described using the equation: x_A = 10t, where x_A represents the distance of ship A from its starting point and t represents time in hours.

Similarly, ship B can be described using the equation: y_B = 100 - 10t, where y_B represents the distance of ship B from its starting point.

Calculating the distance between the ships:
The distance between the two ships can be calculated using the distance formula: distance = sqrt((x_A - x_B)^2 + (y_A - y_B)^2).

Substituting the values of x_A and y_B from the above equations, we get: distance = sqrt((10t)^2 + (100 - 10t)^2).

Finding the time when the distance is shortest:
To find the time when the distance between the ships is shortest, we need to minimize the distance equation. We can do this by differentiating the equation with respect to time and equating it to zero.

Differentiating the distance equation with respect to t, we get:
d(distance)/dt = 0
Simplifying, we have:
2(10t)(10) - 2(100 - 10t)(10) = 0
Solving this equation, we can find the value of t when the distance is shortest.

Calculating the time:
Expanding the equation and simplifying, we get:
200t - 2000 + 200t = 0
400t - 2000 = 0
400t = 2000
t = 5

Therefore, the time after which the distance between the two ships becomes shortest is 5 hours.