A ship sails due west at a speed of 39 km/h. In a direction of 30 degrees to P, what is the resultant velocity of the ship P? Also, relate it to the current due east at 6 km/h. Use... A ship sails due west at a speed of 39 km/h. In a direction of 30 degrees to P, what is the resultant velocity of the ship P? Also, relate it to the current due east at 6 km/h. Use the necessary equations to compute the resulting vector and angles.
Understand the Problem
The question is asking for calculations and relationships regarding a ship's current, sail settings, and distances. It references specific values and relationships to determine resultant vectors, likely within a navigational context.
Answer
The resultant velocity of ship P is 33 km/h at 180 degrees.
Answer for screen readers
The resultant velocity of ship P is 33 km/h in a direction of 180 degrees.
Steps to Solve
- Identify the Velocity Components of the Ship
The ship is moving due west at a speed of 39 km/h. The velocity vector can be represented as: $$ \mathbf{V}_{\text{ship}} = (-39, 0) $$ This means the ship's velocity in the x-direction (west) is -39 km/h, and in the y-direction (north) is 0 km/h.
- Calculate the Current's Velocity Vector
The current's speed is given as 6 km/h, moving due east. Therefore, its velocity vector is: $$ \mathbf{V}_{\text{current}} = (6, 0) $$ This indicates a velocity of 6 km/h in the positive x-direction (east).
- Combine the Velocity Vectors
To find the resultant velocity of the ship (P), we need to sum the two vectors: $$ \mathbf{V}{\text{resultant}} = \mathbf{V}{\text{ship}} + \mathbf{V}{\text{current}} $$ Substituting the values: $$ \mathbf{V}{\text{resultant}} = (-39, 0) + (6, 0) = (-39 + 6, 0) = (-33, 0) $$
- Determine the Magnitude of the Resultant Velocity
The magnitude of the resultant velocity can be calculated using: $$ |\mathbf{V}_{\text{resultant}}| = \sqrt{(-33)^2 + 0^2} = \sqrt{1089} = 33 \text{ km/h} $$
- Find the Direction of the Resultant Velocity
Since the resultant velocity points directly along the negative x-axis, the direction angle relative to the east (positive x-direction) is: $$ \theta = 180^\circ $$
The resultant velocity of ship P is 33 km/h in a direction of 180 degrees.
More Information
The ship’s velocity accounts for both its own speed and that of the eastward current, leading to a net speed heading westward. In navigation, it’s essential to consider both the ship's motion and external factors like currents for accurate course plotting.
Tips
- Misinterpreting Directions: Failing to correctly denote west as negative and east as positive can lead to incorrect vector additions. Always keep track of the direction of each velocity vector.
- Forgetting to Include Components: Ensure each vector's components are considered when adding them together. Missing a component can lead to errors in resultant calculations.
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