Two ships start sailing from a port with an initial distance of 'd' between them with the same constant velocity of 'v' each. First ship sails perpendicular to the coastline and the velocity vector of the second ship is always directed towards the first ship with the same magnitude of 'v'. What is the final constant distance between the two ships?
Let's analyze the situation step by step:
Step 1: Initially, the two ships are at a distance of 'd' from each other.
Step 2: The first ship sails perpendicular to the coastline. This means its direction is at a right angle to the direction of the second ship.
Step 3: The second ship always moves towards the first ship with a constant velocity of 'v'. Since the velocity vector is directed towards the first ship, it implies that the second ship is continuously adjusting its course to keep moving towards the first ship.
Step 4: Since the first ship is moving perpendicular to the coastline, and the second ship is continually adjusting its course towards the first ship, the two ships will eventually meet.
Step 5: When the two ships meet, the final distance between them will be zero. In other words, they will be at the same position.
Therefore, the final constant distance between the two ships will be zero.
To find the final constant distance between the two ships, we can analyze their motion and determine when they will be at their closest point to each other.
Let's assume that the first ship starts at the origin (0, 0), sailing perpendicular to the coastline (along the x-axis) with a constant velocity of 'v'.
The second ship's velocity vector is always directed towards the first ship, with the same magnitude of 'v'. This means that the second ship will move along a line that connects it to the first ship.
Now, let's consider the position of the first ship at any given time 't'. Since the ship is sailing perpendicular to the coastline, its position can be described as (vt, 0).
The position of the second ship at the same time 't' can be found by using the position vector connecting the two ships. Since the second ship's velocity vector is directed towards the first ship, the position vector will have the same direction. Additionally, we know that the magnitude of the velocity vector for the second ship is 'v', so the position vector will have a magnitude of 'vt'.
Therefore, the position of the second ship at time 't' can be described as (vt, vt).
Now, let's calculate the distance between the two ships at time 't' using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the positions of the ships at time 't':
Distance = √(((vt - 0)^2) + ((vt - 0)^2))
Simplifying the equation:
Distance = √(2v^2t^2)
Taking the square root of 2v^2, we get:
Distance = vt√2
Since both ships have the same constant velocity 'v', the distance between them at any time 't' is proportional to 't', and therefore, is also constant.
So, the final constant distance between the two ships is vt√2, where 'v' is the constant velocity and 't' is the time.