a ship is 100km north west of cape town harbour and it sails directly north at a velocity of 30km/h in ei

ght hours.calculate its resultant speed with reference to the cape town harbour in magnitude and direction

To calculate the resultant speed with reference to Cape Town Harbour, we need to use vector addition.

First, let's analyze the given information:
1. The ship is initially 100 km northwest of Cape Town Harbour.
2. It sails directly north at a velocity of 30 km/h for 8 hours.

To calculate the resultant speed, we will use the Pythagorean theorem and trigonometry.

Step 1: Find the displacement vector of the ship's northward motion
The displacement vector is calculated by multiplying the velocity by the time:
Displacement = Velocity * Time
Displacement = 30 km/h * 8 h = 240 km north

Step 2: Find the displacement vector of the ship's northwestward position
Since the ship is initially located 100 km northwest of Cape Town Harbour, the displacement vector is 100 km northwest.

Step 3: Calculate the resultant displacement vector
To find the resultant displacement, we need to add the northward and northwestward displacement vectors using vector addition.

Let's represent the northward displacement as vector A (240 km north) and the northwestward displacement as vector B (100 km northwest).

The resultant displacement vector, vector R, is given by:
R = A + B

To add these vectors, we can break them down into their horizontal (x) and vertical (y) components.

Vector A: 240 km north
Ax = 0 km (no horizontal component)
Ay = 240 km (vertical component)

Vector B: 100 km northwest
Bx = -100 km (horizontal component)
By = 100 km (vertical component)

Summing the x and y components separately:

Rx = Ax + Bx = 0 km + (-100 km) = -100 km (horizontal component)
Ry = Ay + By = 240 km + 100 km = 340 km (vertical component)

Using the Pythagorean theorem, we can calculate the magnitude (magnitude R) of the resultant displacement:

magnitude R = √(Rx^2 + Ry^2)
magnitude R = √((-100 km)^2 + (340 km)^2)
magnitude R = √(10000 km^2 + 115600 km^2)
magnitude R = √125600 km^2
magnitude R ≈ 354.68 km

Step 4: Calculate the direction of the resultant displacement
The direction can be determined by finding the angle the resultant vector makes with the north direction. We can use trigonometry to calculate this angle.

angle = arctan (Ry / Rx)
angle = arctan (340 km / -100 km)

To find the angle, we need to determine the quadrant:

Quadrant: Since Rx is negative (-100 km) and Ry is positive (340 km), the angle will be in the second quadrant.

To find the angle, we can calculate the inverse tangent (arctan) of the ratio (∣Ry / Rx∣), and then add 180 degrees to get the angle in the second quadrant.

angle = arctan (∣Ry / Rx∣) + 180 degrees
angle = arctan (∣340 km / -100 km∣) + 180 degrees

You can use a calculator to find the exact value of the angle in degrees.

Therefore, the magnitude of the resultant speed is approximately 354.68 km, and the direction is determined by the angle we calculated.