A sailboat is heading directly north at a speed of 16 knots (1 knot = 0.514 m/s). The wind is blowing towards the east with a speed of 10.4 knots. (Both velocities are given relative to the Earth.)
Determine the magnitude and direction of the wind velocity as measured on the boat.
north wind due to boat speed at 16 kn
west wind at 10.4 kn
magnitude = sqrt(16^2+10.4^2)
tan angle west of north = 10.4/16
compass angle= 360 - that angle
To determine the magnitude and direction of the wind velocity as measured on the boat, we need to combine the velocity of the boat (heading north) and the velocity of the wind (blowing east) using vector addition.
Step 1: Convert the given velocities to meters per second (m/s).
- Boat's velocity: 16 knots = 16 * 0.514 m/s = 8.224 m/s
- Wind's velocity: 10.4 knots = 10.4 * 0.514 m/s = 5.3376 m/s
Step 2: Represent the boat's velocity as a vector.
Since the boat is heading directly north, the boat's velocity vector can be represented as (0 m/s, 8.224 m/s).
Step 3: Represent the wind's velocity as a vector.
Since the wind is blowing towards the east, the wind's velocity vector can be represented as (5.3376 m/s, 0 m/s).
Step 4: Add the boat's velocity vector and the wind's velocity vector to get the relative velocity of the wind as measured on the boat.
(0 m/s, 8.224 m/s) + (5.3376 m/s, 0 m/s) = (5.3376 m/s, 8.224 m/s)
Step 5: Determine the magnitude (speed) and direction of the wind velocity on the boat.
Magnitude: The magnitude of the wind velocity vector on the boat can be calculated using the Pythagorean theorem, ||v|| = sqrt(Vx^2 + Vy^2), where Vx and Vy represent the vector components.
||v|| = sqrt((5.3376 m/s)^2 + (8.224 m/s)^2) = sqrt(28.50715 + 67.655776) = sqrt(96.162926) = 9.805 m/s
Direction: The direction of the wind velocity vector on the boat can be calculated using the inverse tangent function, atan2(Vy, Vx).
Direction = atan2(8.224 m/s, 5.3376 m/s) = 56.38 degrees
Therefore, the magnitude of the wind velocity as measured on the boat is approximately 9.805 m/s, and the direction is approximately 56.38 degrees relative to the north on the boat.