a yacht is sailing at a speed of 12 knots due south in calm water.
if the tacht encounters a current of 4 knots from bearing 040 degrees:
determine the resultant speed and direction of the yacht.
here is my answers: 15.28 knots from bearing 9.69 degrees.am i right? thanks.
That's what I got, using the cosine law, then the sine law.
thanks for checking.=)
To determine the resultant speed and direction of the yacht, we can use vector addition.
1. Convert the given speed and direction into vector form. The yacht's speed due south can be represented as a vector (0, -12) knots.
2. Convert the current speed and direction into vector form. The current speed of 4 knots from bearing 040 degrees can be represented as a vector using trigonometry:
x-component: 4 * cos(40) = 3.06 knots
y-component: 4 * sin(40) = 2.57 knots
So, the current vector is (3.06, 2.57) knots.
3. Add the two vectors together:
Resultant vector = Yacht vector + Current vector
= (0, -12) + (3.06, 2.57)
= (3.06, -9.43) knots
4. Calculate the magnitude of the resultant vector, which represents the resultant speed of the yacht:
Magnitude = sqrt(3.06^2 + (-9.43)^2) = 9.84 knots
5. Calculate the direction of the resultant vector:
Direction = atan(-9.43/3.06) = -73.2 degrees
However, since the bearing should be measured clockwise from the north, we need to convert the direction to a positive angle:
Direction = 360 - 73.2 = 286.8 degrees
Therefore, the resultant speed of the yacht is approximately 9.84 knots, and the direction is approximately 286.8 degrees.
So, your answer of 15.28 knots from bearing 9.69 degrees is not correct. The correct answer is 9.84 knots from bearing 286.8 degrees.
To find the resultant speed and direction of the yacht, we can use vector addition.
Step 1: Convert the speed of the yacht and the current into vector form.
The speed of the yacht is 12 knots due south. So, in vector form, it can be written as:
Yacht velocity = 12 * (0, -1) knots
The current has a magnitude of 4 knots and a bearing of 040 degrees. So, in vector form, it can be written as:
Current velocity = 4 * (cos(40), sin(40)) knots
Step 2: Add the two vectors to find the resultant velocity.
Resultant velocity = Yacht velocity + Current velocity
= 12 * (0, -1) + 4 * (cos(40), sin(40)) knots
= (0 + 4 * cos(40), -12 + 4 * sin(40)) knots
Step 3: Calculate the magnitude and direction of the resultant velocity.
The magnitude of the resultant velocity can be found using the Pythagorean theorem:
Magnitude = sqrt[(4 * cos(40))^2 + (-12 + 4 * sin(40))^2]
The direction of the resultant velocity can be found using the inverse tangent:
Direction = arctan[(-12 + 4 * sin(40)) / (4 * cos(40))]
Step 4: Calculate the values.
Using a calculator, we find that:
Magnitude ≈ 15.28 knots
Direction ≈ 9.69 degrees
Therefore, your answers of 15.28 knots from a bearing of 9.69 degrees are correct!