A ship is heading north at 15km/h in a tide moving at 6km/h westward. 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, we can use vector addition.
First, let's define the directions:
North will be considered as the positive y-axis, and west will be considered as the negative x-axis.
Given:
Ship's velocity northward = 15 km/h
Tide's velocity westward = 6 km/h
Since we are dealing with velocities, we need to represent them as vectors.
The velocity of the ship heading northward can be represented as V_ship = 15 km/h in the positive y direction (northward).
The velocity of the tide moving westward can be represented as V_tide = 6 km/h in the negative x direction (westward).
To find the resultant velocity, we need to add these two velocity vectors.
Since the vectors are at right angles to each other, we can use the Pythagorean theorem to find the magnitude of the resultant velocity:
Magnitude of the resultant velocity (V_resultant) = sqrt[V_ship^2 + V_tide^2]
Substituting the given values:
V_resultant = sqrt[(15 km/h)^2 + (6 km/h)^2]
V_resultant = sqrt[225 km^2/h^2 + 36 km^2/h^2]
V_resultant = sqrt[261 km^2/h^2]
V_resultant ≈ 16.14 km/h (approximated to two decimal places)
The magnitude of the resultant velocity is approximately 16.14 km/h.
To determine the direction of the resultant velocity, we can use trigonometry. Since the ship is heading north and the tide is moving west, the resultant velocity will form a right triangle with the positive x-axis.
To find the direction, we can use the tangent of the angle between the resultant velocity and the positive x-axis:
tan(θ) = V_ship / V_tide
θ ≈ arctan(V_ship / V_tide)
θ ≈ arctan(15 km/h / 6 km/h)
θ ≈ arctan(2.5)
Using a calculator, the approximate angle is θ ≈ 68.2 degrees.
Therefore, the magnitude of the resultant velocity is 16.14 km/h, and its direction is approximately 68.2 degrees north of west.