A ship us heading north at 15 km/h in a tide moving at 6 km/h west ward. Determine the magnitude and direction of the resultant velocity of the ship.
To determine the magnitude and direction of the resultant velocity of the ship, you can use vector addition.
First, let's represent the northward velocity of the ship as a vector. Since the ship is heading north at a speed of 15 km/h, we can represent it as a vector of magnitude 15 km/h pointing in the north direction.
Next, let's represent the westward velocity of the tide as another vector. Since the tide is moving at a speed of 6 km/h westward, we can represent it as a vector of magnitude 6 km/h pointing in the west direction.
To add the two vectors, place the tail of the second vector at the head of the first vector. In this case, we place the tail of the westward tide vector at the head of the northward ship vector.
Now, draw the resultant vector from the tail of the first vector (northward ship) to the head of the second vector (westward tide). The resultant vector represents the magnitude and direction of the resultant velocity.
To determine the magnitude of the resultant velocity, we can use the Pythagorean theorem. The magnitude of the resultant velocity can be calculated as follows:
Resultant velocity = √(northward velocity^2 + westward velocity^2)
= √(15^2 + 6^2)
= √(225 + 36)
= √261
≈ 16.155 km/h
The direction of the resultant velocity can be determined by finding the angle between the resultant and the reference direction (north). To find this angle, we can use trigonometry. We have a right-angled triangle, where the northward velocity is the opposite side and the westward velocity is the adjacent side.
The angle θ can be calculated as follows:
θ = tan^(-1) (opposite/adjacent)
= tan^(-1) (15/6)
= tan^(-1) (2.5)
≈ 68.2 degrees
Therefore, the magnitude of the resultant velocity is approximately 16.155 km/h, and the direction is approximately 68.2 degrees north of west.