A ship's course is set at a heading of 192 degrees, with a speed of 30 knots. A current is flowung from a bearing of 112 degrees, at 14 knots. Use Cartesian vectors to determine the resultant velocity of the ship.
Perform a vector addition of the ship's velocity with respect to the water (V1), plus the velocity of the water with respect to land (V2).
i = unit vector east
j = unit vector north
V1 = 30 sin 192 i + 30 cos 192 j
V2 = 14 sin 112 i + 14 cos 112 j.
V1 + V2 = ?
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To determine the resultant velocity of the ship using Cartesian vectors, we need to break down the velocities into their x and y components.
Let's start with the velocity of the ship.
The ship's heading of 192 degrees can be visualized as an angle measured counterclockwise from the positive x-axis. To find the x and y components of the ship's velocity, we can use trigonometry.
The x-component of the ship's velocity can be calculated as:
Vx_ship = speed_ship * cos(heading_ship)
The y-component of the ship's velocity can be calculated as:
Vy_ship = speed_ship * sin(heading_ship)
Now, let's calculate the x and y components of the ship's velocity.
Vx_ship = 30 knots * cos(192 degrees)
≈ -22.79 knots
Vy_ship = 30 knots * sin(192 degrees)
≈ -15.25 knots
Next, let's determine the velocity of the current.
The current's bearing of 112 degrees can also be visualized as an angle measured counterclockwise from the positive x-axis.
Vx_current = speed_current * cos(bearing_current)
Vy_current = speed_current * sin(bearing_current)
Calculating the x and y components of the current's velocity:
Vx_current = 14 knots * cos(112 degrees)
≈ -10.25 knots
Vy_current = 14 knots * sin(112 degrees)
≈ 10.55 knots
To find the resultant velocity, we need to add the x and y components of the ship's velocity to the x and y components of the current's velocity.
Vx_resultant = Vx_ship + Vx_current
≈ -22.79 knots - 10.25 knots
≈ -33.04 knots
Vy_resultant = Vy_ship + Vy_current
≈ -15.25 knots + 10.55 knots
≈ -4.70 knots
Finally, we can use the Pythagorean theorem to find the magnitude (speed) of the resultant velocity:
Resultant speed = √(Vx_resultant^2 + Vy_resultant^2)
= √((-33.04 knots)^2 + (-4.70 knots)^2)
≈ 33.37 knots
Therefore, the resultant velocity of the ship is approximately 33.37 knots, in a direction of approximately 254.94 degrees counterclockwise from the positive x-axis.
To determine the resultant velocity of the ship using Cartesian vectors, we first need to break down the given velocities (ship's speed and current's speed) into their x and y components.
Let's start with the ship's velocity. The heading of 192 degrees can be converted to a unit vector by using cosine and sine functions. The x-component (vship_x) and y-component (vship_y) of the ship's velocity can be calculated as follows:
vship_x = magnitude of ship's speed * cos(heading)
= 30 knots * cos(192 degrees)
≈ -18.634 knots
vship_y = magnitude of ship's speed * sin(heading)
= 30 knots * sin(192 degrees)
≈ -26.802 knots
Next, let's calculate the current's velocity components. The bearing of 112 degrees can also be converted to a unit vector by using cosine and sine functions. The x-component (vcurrent_x) and y-component (vcurrent_y) of the current's velocity can be calculated as follows:
vcurrent_x = magnitude of current's speed * cos(bearing)
= 14 knots * cos(112 degrees)
≈ -6.642 knots
vcurrent_y = magnitude of current's speed * sin(bearing)
= 14 knots * sin(112 degrees)
≈ 11.208 knots
Now, we can calculate the resultant velocity (vresultant) by summing up the x and y components of the ship's and current's velocities:
vresultant_x = vship_x + vcurrent_x
≈ -18.634 knots + (-6.642 knots)
≈ -25.276 knots
vresultant_y = vship_y + vcurrent_y
≈ -26.802 knots + 11.208 knots
≈ -15.594 knots
Therefore, the resultant velocity of the ship is approximately -25.276 knots in the x-direction and -15.594 knots in the y-direction.