A ship is sailing due north at 12km/h while another ship is observed 15km ahead, travellers due east at 9 km/h. What is the closest distance of approach of the two ships?
To find the closest distance of approach between the two ships, we can use the concept of relative motion.
Step 1: Determine the velocity vectors of both ships. Let's call the ship sailing due north Ship A, and the ship traveling due east Ship B. Ship A has a velocity vector of 12 km/h due north, and Ship B has a velocity vector of 9 km/h due east.
Step 2: Convert the velocity vectors into Cartesian coordinates. The velocity vector of Ship A can be written as (0, 12) km/h, where the first component represents the horizontal (east-west) velocity and the second component represents the vertical (north-south) velocity. The velocity vector of Ship B can be written as (9, 0) km/h.
Step 3: Calculate the relative velocity vector. To do this, subtract the two velocity vectors. The relative velocity vector is obtained by subtracting the velocity vector of Ship B from the velocity vector of Ship A: (0, 12) - (9, 0) = (-9, 12) km/h.
Step 4: Find the magnitude of the relative velocity vector. The magnitude of a vector is the length or size of the vector. For the relative velocity vector, the magnitude is calculated using the Pythagorean theorem: Magnitude = sqrt((-9)^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 km/h.
Step 5: The closest distance of approach is given by the formula Distance = Relative Velocity * Time. Since we are interested in the closest distance of approach, we assume that the two ships are on a collision course, and so the time of closest approach is zero. Therefore, the closest distance of approach is 15 km/h * 0 = 0 km.
Therefore, the closest distance of approach between the two ships is zero, indicating that they will collide.