Help with this problem please. I don't get it at all.
A ship is heading due north at 12 mph. The current is flowing southwest at 4mph. Find the actual bearing and speed of the ship.
Are you studying vectors?
I see a triangle with triangle ABC with AB vertical with length 12, angle B = 45 degrees, and BC = 4
by cosine law:
b^2 = 4^2 + 12^2 - 2(4)(12)cos45
= 92.1177
b = 9.598
Sos the actual speed of the ship is 9.6 mph
Use the Sine Law to find angle A for the bearing
(I had 17.14 degrees)
The bearing would be 360-17.14= 342.86 due to the fact that the bearing scale is different than the unit circle.
To find the actual bearing and speed of the ship, we can use vector addition.
Step 1: Draw a diagram
Draw a diagram to represent the situation. Let's use an arrow to represent the ship's velocity due north and another arrow to represent the current's velocity flowing southwest.
Step 2: Decompose the velocities
Decompose the ship's velocity and the current's velocity into their x and y components.
The ship's velocity due north is 12 mph in the y-direction (positive).
The current's velocity flowing southwest can be decomposed into components: -4 mph in the x-direction (negative) and -4 mph in the y-direction (negative).
Step 3: Determine the resultant velocity
To find the resultant velocity, we need to add the ship's velocity vector and the current's velocity vector.
The x-component of the resultant velocity is the sum of the x-components of the individual velocities: 0 mph + (-4 mph) = -4 mph.
The y-component of the resultant velocity is the sum of the y-components of the individual velocities: 12 mph + (-4 mph) = 8 mph.
Step 4: Calculate the magnitude and direction
The magnitude of the resultant velocity can be found using the Pythagorean theorem:
Magnitude = sqrt((-4 mph)^2 + (8 mph)^2) ≈ 8.94 mph.
To determine the direction, we can use trigonometry. The angle θ between the resultant velocity vector and the positive x-axis can be found using the inverse tangent (arctan) function:
θ = arctan(8 mph / -4 mph) ≈ -63.4°.
Step 5: Convert the direction to compass bearing
To convert the direction to compass bearing, we need to add 90° to the angle θ and adjust for the quadrant.
Compass bearing = 270° - 63.4° = 206.6° (rounded to one decimal place).
Therefore, the actual bearing and speed of the ship is approximately 206.6° at 8.94 mph.
To find the actual bearing and speed of the ship, we need to use vector addition. Let's break down the velocity of the ship and the velocity of the current into their respective components.
The ship is heading due north at 12 mph. Since it's heading due north, the velocity can be represented as (0, 12), where the first component represents the velocity in the east-west direction (x-axis) and the second component represents the velocity in the north-south direction (y-axis).
The current is flowing southwest at 4 mph. The angle southwest makes with the positive x-axis is 45 degrees (since southwest is the combination of south and west which makes a 45-degree angle with the positive x-axis in the counter-clockwise direction). To find the x and y components of the current, we can use the sine and cosine functions.
cos(45°) = sin(45°) = √2 / 2
So, the x component of the current is 4 mph × (√2 / 2) = 2√2 mph, and the y component of the current is -4 mph × (√2 / 2) = -2√2 mph (negative because it's flowing in the opposite direction).
Now, let's add the x and y components of the ship's velocity and the current's velocity:
x component: 0 + 2√2 = 2√2 mph
y component: 12 - 2√2 = 12 - 2√2 mph
To find the magnitude of the resulting velocity (speed) and the direction (bearing), we can use the Pythagorean theorem and trigonometry. The magnitude can be calculated as:
magnitude = √(x component^2 + y component^2)
magnitude = √((2√2)^2 + (12 - 2√2)^2)
magnitude = √(8 + 144 - 48√2 + 8)
magnitude = √(160 - 48√2)
magnitude ≈ 12.92 mph
To find the direction (bearing), we can calculate the angle the resulting velocity makes with the positive x-axis (east). We can use the inverse tangent (arctan) function:
angle = arctan(y component / x component)
angle = arctan((12 - 2√2) / 2√2)
Now, we can use a calculator to find the value of the angle in degrees or approximate it using decimal places.
angle ≈ 73.74 degrees
Therefore, the ship's speed is approximately 12.92 mph, and its bearing (direction) is approximately 73.74 degrees.