At the entrance channel of a harbor, the tidal current has a velocity of 4.94 km/hr in a direction 23.2° south of east. Suppose a ship caught in this current has a speed of 15.6 km/hr relative to the water. If the helmsman keeps the bow of the ship aimed north, what will be the speed of the ship relative to the ground?
I know this is vector addition, but I'm unsure on which steps I need to take first.
Break the tidal current into components, S and E. Add that to the N movement of the ship.
Then, put the N and E components together to get the answer. I will be happy to critique your thinking.
To find the speed of the ship relative to the ground, we need to add the velocity of the tidal current (4.94 km/hr at 23.2° south of east) to the velocity of the ship relative to the water (15.6 km/hr aimed north).
Here are the steps to find the speed of the ship relative to the ground:
Step 1: Convert the given velocities to vector form.
- The velocity of the tidal current (4.94 km/hr) at 23.2° south of east can be represented as a vector: 4.94 km/hr at 23.2° south of east.
- The velocity of the ship relative to the water (15.6 km/hr) aimed north can be represented as a vector: 15.6 km/hr aimed north.
Step 2: Resolve the vectors into their components.
- The velocity of the tidal current can be resolved into its north and east components using trigonometry. The north component (Vy) can be found by multiplying the magnitude of the vector (4.94 km/hr) by the sine of the angle (23.2°). Similarly, the east component (Vx) can be found by multiplying the magnitude by the cosine of the angle.
Vy = 4.94 km/hr * sin(23.2°)
Vx = 4.94 km/hr * cos(23.2°)
Step 3: Add the components of the velocities.
- Add the north components (Vy) of both vectors to find the total north component.
- Add the east components (Vx) of both vectors to find the total east component.
- This will give us the total velocity vector of the ship relative to the ground.
Step 4: Calculate the magnitude and direction of the total velocity vector.
- Use the Pythagorean theorem to find the magnitude (speed) of the total velocity vector.
- Use trigonometry to find the direction of the total velocity vector.
Step 5: Ensure the direction is in a standard notation.
- Convert the direction to a standard notation, such as in terms of north and east or using angles relative to a reference direction (e.g., north or east).
By following these steps, you should be able to find the speed of the ship relative to the ground.