A ship is 100km north-west of cape town harbour and it sails directly north at a velocity of 30km/h in 8 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 find the magnitude and direction of the ship's displacement.
First, let's find the ship's displacement after 8 hours of sailing at 30 km/h north. Since the ship is sailing directly north, its displacement in the north direction will be given by:
Displacement_North = Velocity_North * Time = 30 km/h * 8 hours = 240 km
Now, let's consider the initial position of the ship. It is 100 km northwest of Cape Town Harbour. Northwest is a combination of north and west. Since the ship is traveling north, we can disregard the westward distance.
Therefore, the displacement in the north direction from the initial position is:
Displacement_Northwest = Sin(45°) * 100 km = √2/2 * 100 km ≈ 70.7 km
Next, we use the Pythagorean theorem to calculate the magnitude of the resultant displacement:
Resultant_Displacement = √(Displacement_North² + Displacement_Northwest²)
= √(240 km)² + (70.7 km)²)
= √(57,600 km² + 5,000 km²)
= √62,600 km²
≈ 250 km
Finally, we need to determine the direction of the resultant displacement, which can be found using trigonometry:
Resultant_Direction = arctan(Displacement_Northwest / Displacement_North)
= arctan(70.7 km / 240 km)
≈ 16.7°
Therefore, the resultant speed with reference to Cape Town Harbour is approximately 250 km in magnitude and 16.7° north of east in direction.