A freighter, streaming on course 140„a at 20 knots, is 40 nautical miles N20„aE of a submarine with a cruising speed of 25 knots. Find the course to be set by the sub to overtake the freighter in the least amount of time, and find this minimum time.
Let's place the submarine at (0,0) to start.
The freighter is 40 miles at N20°E, placing it at (13.68,37.59)
The freighter's heading is 140°, so at time t its position is
(fx,fy) = (13.68 + 20t*cos(140°), 37.59 + 20t*sin(140°))
(fx,fy) = (13.68-15.32t , 37.59 + 12.86t)
The sub, at speed 25, sails at a heading of a°, so its position at time t is
(sx,sy) = (25t*cos(a),25t*sin(a))
Now we can figure the distance between the vessels:
d^2 = (fx-sx)^2 + (fy-sy)^2
The equation is in two variables, and is not subject to direct solution, but a little iteration will show that the two ships will meet at
time t = 4.15 hours if the sub's heading is 118.75° or N28.75°W.
Hmmm. It occurs to me that I have been using common trig angles, with 0° pointing due East. Nautical headings have 0° = due North, so you may have to fiddle with the angles to correct things.
Thinking about things a bit more, I realized that a simple law of cosines setup would work.
(25t)^2 = 40^2 + (20t)^2 - 2*40*cos110°
625t^2 = 1600 + 400t^2 + 684t
225t^2 - 684t - 1600 = 0
t = 4.58 (a little different value)
20t = 91.6
25t = 114.5
sin(a)/91.6 = sin110°/114.5
sin(a) = .75175
a = 48.74°
add that to the 70° initial bearing to get 118.74°
To find the course to be set by the submarine to overtake the freighter in the least amount of time, we need to calculate the relative course between the two vessels.
1. Start by drawing a diagram to represent the positions and courses of the freighter and the submarine. Here's a simple representation:
F (140°)
/
/
/
S (?????)
F = Freighter
S = Submarine
2. Calculate the initial relative bearing between the two vessels. We can do this by subtracting the course of the submarine from the course of the freighter. In this case, it would be: 140° - 180° = -40°.
The negative sign indicates that the submarine is to the left of the freighter when facing its original course.
3. Next, calculate the true heading difference (THD) between the two vessels. The formula to calculate THD is: THD = 180° - |Relative Bearing|. In this case, it would be: THD = 180° - |-40°| = 180° - 40° = 140°.
4. Now, calculate the time it takes for the submarine to overtake the freighter. The time can be calculated using the following formula: Time = Distance / Speed. In this case, the distance is 40 nautical miles, and the speed of the submarine is 25 knots. So the time taken for the submarine to overtake the freighter is: Time = 40 / 25 = 1.6 hours.
5. Finally, we need to calculate the course for the submarine to overtake the freighter in the least amount of time, which is equal to the true heading difference (THD). In this case, the course to be set by the submarine would be: 140°.
To summarize:
- The course to be set by the submarine to overtake the freighter in the least amount of time is 140°.
- The minimum time required for the submarine to overtake the freighter is 1.6 hours.