A ship leaves port at noon at heads due east at 20 nautical miles/hour (20 knots). At 2PM the ship changes course to N 54° W.
From the port of departure towards the ship at 3 PM, find the following:
a) the bearing to the ship (to the nearest degree)
b) the distance to the ship (round to 1 decimal place)
Where is the ship with respect to the starting point (0,0)
I assume we do 20 knots the whole time.
East 40 - 20 sin 54 = 40 - 16.2 = 23.8 East of start
North 20 cos 54 = 11.8 North of start
tan angle East of North = 23.8/11.8
bearing angle East of North = 63.6 deg. Note that is compass bearing, not on math xy system
distance = sqrt (11.8^2 + 23.8^2) nautical mile
a) Well, if the ship changed course at 2 PM, it's probably still trying to find its way. Let's call it the "Lost Ship" for now! So, since it's 3 PM now, the bearing to the Lost Ship is on its GPS, trying to navigate back to the correct course. But let's just say it's probably somewhere between "Are you sure this is North?" and "Hey, let's ask for directions!" degrees.
b) Ah, distance. The Lost Ship is probably thinking, "If I just go a little further, maybe I'll find a map!" So, let's see. The ship has been sailing east at 20 knots for 2 hours and then changed direction. So, it has covered a total distance of 20 knots/hour * 2 hours = 40 nautical miles. But since it changed direction, the actual distance to the ship depends on how far it went after changing course. Looks like we have to wait for the next GPS update to find out!
To find the bearing and distance to the ship, we can break down the problem into several steps:
Step 1: Determine the ship's position at 2 PM
Since the ship travels east at a constant speed of 20 knots for 2 hours from noon, we can calculate its position at 2 PM. The distance covered is given by:
Distance = Speed × Time
Distance = 20 knots × 2 hours
Distance = 40 nautical miles
Therefore, at 2 PM, the ship is 40 nautical miles directly east of the port.
Step 2: Determine the ship's position at 3 PM
From 2 PM to 3 PM, the ship changes course to N 54° W. This means the ship is traveling in a direction that is 54 degrees west of north. Since we know the speed of the ship is 20 knots, we can calculate the distance traveled in one hour.
To find the eastward component of the distance traveled, we use trigonometry:
Eastward Distance = Distance × cos(angle)
Eastward Distance = 20 knots × cos(54°)
Eastward Distance ≈ 20 knots × 0.5878 ≈ 11.76 nautical miles
To find the northward component of the distance traveled, we use trigonometry again:
Northward Distance = Distance × sin(angle)
Northward Distance = 20 knots × sin(54°)
Northward Distance ≈ 20 knots × 0.8090 ≈ 16.18 nautical miles
Adding these two components, the ship has traveled approximately 11.76 nautical miles east and 16.18 nautical miles north in one hour.
Step 3: Calculate the ship's position at 3 PM
To determine the ship's position at 3 PM, we add the distance traveled in one hour to its position at 2 PM:
Eastward Position = 40 nautical miles + 11.76 nautical miles ≈ 51.76 nautical miles east
Northward Position = 0 nautical miles + 16.18 nautical miles ≈ 16.18 nautical miles north
At 3 PM, the ship is approximately 51.76 nautical miles east and 16.18 nautical miles north of the port.
Step 4: Calculate the bearing to the ship at 3 PM
To find the bearing to the ship from the port at 3 PM, we use trigonometry again. The bearing represents the angle between the north direction and the line connecting the port and the ship's position.
First, we calculate the difference in eastward and northward distances between the port and the ship at 3 PM:
Eastward Difference = Eastward Position - Port's Eastward Position = 51.76 nautical miles - 0 nautical miles = 51.76 nautical miles
Northward Difference = Northward Position - Port's Northward Position = 16.18 nautical miles - 0 nautical miles = 16.18 nautical miles
Next, we calculate the bearing using the arctan function (inverse tangent):
Bearing = arctan(Eastward Difference / Northward Difference)
Bearing = arctan(51.76 / 16.18) ≈ 72.32°
Therefore, the bearing to the ship at 3 PM is approximately 72 degrees.
Step 5: Calculate the distance to the ship at 3 PM
To find the distance to the ship at 3 PM, we use the distance formula (Pythagorean theorem):
Distance^2 = (Eastward Difference)^2 + (Northward Difference)^2
Distance^2 = (51.76)^2 + (16.18)^2
Distance ≈ √(2677.5376 + 262.0324) ≈ √2939.57 ≈ 54.2 nautical miles
Therefore, the distance to the ship at 3 PM is approximately 54.2 nautical miles.