A ship travels NE (45 degrees) at 18 knots with a current of 8 knots direction North. What is the bearing of the ship and speed in knots?
18@NE = (12.73,12.73)
8@N = (0,8)
add them up to get the net speed: (12.73,20.73)
The resultant heading is
24.32 knots @ E58.75°N
To find the bearing and speed of the ship, we need to solve this problem step-by-step.
Step 1: Determine the horizontal and vertical components of the ship's velocity.
The ship is traveling at 18 knots in a direction 45 degrees NE. To find the horizontal and vertical components, we can use trigonometry.
Horizontal component = 18 knots * cos(45°)
Vertical component = 18 knots * sin(45°)
Using a calculator, we find:
Horizontal component = 12.73 knots (rounded to two decimal places)
Vertical component = 12.73 knots (rounded to two decimal places)
Step 2: Determine the horizontal and vertical components of the current.
The current has a velocity of 8 knots directed North. Since it is directed straight North, the horizontal component is 0, and the vertical component is 8 knots.
Horizontal component of current = 0 knots
Vertical component of current = 8 knots
Step 3: Find the resultant velocity by adding the ship's velocity and the current's velocity.
Horizontal component of resultant velocity = Horizontal component of ship's velocity + Horizontal component of current
Vertical component of resultant velocity = Vertical component of ship's velocity + Vertical component of current
Horizontal component of resultant velocity = 12.73 knots + 0 knots = 12.73 knots
Vertical component of resultant velocity = 12.73 knots + 8 knots = 20.73 knots
Step 4: Find the magnitude and direction of the resultant velocity.
Using the Pythagorean theorem, the magnitude of the resultant velocity is:
Resultant velocity = √(Horizontal component^2 + Vertical component^2)
Resultant velocity = √(12.73^2 + 20.73^2) knots
Resultant velocity = √(162.52 + 430.04) knots
Resultant velocity = √592.56 knots
Resultant velocity ≈ 24.33 knots (rounded to two decimal places)
To find the direction, we can use inverse tangent:
Direction = atan(Vertical component of resultant velocity / Horizontal component of resultant velocity)
Direction = atan(20.73 knots / 12.73 knots)
Direction ≈ 59.97° (rounded to two decimal places)
Since the direction is measured counterclockwise from the East, the bearing of the ship is approximately 59.97° NE (rounded to two decimal places), and the speed is approximately 24.33 knots (rounded to two decimal places).
To determine the bearing of the ship, we need to take into account the effect of both the ship's course (NE at 45 degrees) and the current (North).
First, let's consider the ship's course. NE stands for Northeast, which is a 45-degree angle relative to the North. We can break this angle down into its components using basic trigonometry.
The angle can be split into two equal parts, 22.5 degrees each, representing the ship's movement to the east (E) and north (N). To find the ship's bearing, we need to calculate the tangent of the 22.5-degree angle.
tan(22.5 degrees) = opposite/adjacent = N/E
For the current, which is directly north, we don't need any calculations.
Now, combining the ship's course and the current's direction, we can determine the resultant direction and speed.
Since the ship is traveling at 18 knots NE and the current is 8 knots North, we can use vector addition to determine the resultant speed and direction. We'll treat the ship's speed as the hypotenuse of a right triangle and the current's speed as one of the triangle's legs.
Let's use the Pythagorean theorem to calculate the resultant speed:
Resultant speed^2 = (ship speed)^2 + (current speed)^2
Resultant speed^2 = (18 knots)^2 + (8 knots)^2
Resultant speed^2 = 324 knots^2 + 64 knots^2
Resultant speed^2 = 388 knots^2
Taking the square root of both sides:
Resultant speed = √388 knots
Resultant speed ≈ 19.7 knots
Finally, to determine the bearing, we need to find the angle of the resultant velocity with respect to North. We can use the inverse tangent function (arctan) to calculate this angle:
Bearing = arctan(current speed / ship speed)
Bearing = arctan(8 knots / 18 knots)
Bearing ≈ 25 degrees
Therefore, the ship's bearing is approximately 25 degrees, and its speed is approximately 19.7 knots.