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 sure on which steps I need to take first.
To find the speed of the ship relative to the ground, you need to add the velocities of the tidal current and the ship's velocity. Here are the steps you can follow:
Step 1: Represent the velocities as vectors. Let's denote the tidal current velocity as V_current and the ship's velocity as V_ship.
Step 2: Determine the components of the tidal current velocity. The velocity of 4.94 km/hr at 23.2° south of east can be broken down into its horizontal (x) and vertical (y) components.
V_current_x = V_current * cos(23.2°)
V_current_y = V_current * sin(23.2°)
Step 3: Determine the components of the ship's velocity. Since the ship is aimed north, its velocity will have no horizontal component.
V_ship_x = 0
V_ship_y = V_ship
Step 4: Add the x and y components of the velocities together.
V_ground_x = V_current_x + V_ship_x
V_ground_y = V_current_y + V_ship_y
Step 5: Calculate the magnitude and direction of the ship's velocity relative to the ground. The magnitude (speed) of the ship's velocity can be found using the Pythagorean theorem:
Speed_ground = √(V_ground_x^2 + V_ground_y^2)
The direction of the ship's velocity relative to the ground can be found using the inverse tangent function:
Direction_ground = atan(V_ground_y / V_ground_x)
By following these steps, you'll be able to find the speed and direction of the ship relative to the ground.
To determine the speed of the ship relative to the ground, you need to add the velocity of the current to the velocity of the ship relative to the water. Here are the steps you can follow:
1. Convert the given velocities to their respective vector representations. The velocity of the current is 4.94 km/hr in a direction 23.2° south of east. To represent this as a vector, you can break it down into its eastward and southward components using trigonometry.
- The eastward component = velocity of the current × cos(23.2°)
- The southward component = velocity of the current × sin(23.2°)
2. Determine the components of the velocity of the ship relative to the water. The ship has a velocity of 15.6 km/hr, but since it is aimed north, there is no eastward or southward component. Thus, the eastward and southward components of the ship's velocity relative to the water would be zero.
3. Add the corresponding components together to find the resultant vector. Since the northward component of the ship's velocity is zero, the northward component of the resultant vector will be the same as the northward component of the current's velocity. The eastward component of the resultant vector will be the sum of the eastward components of the current and ship's velocities, and the southward component of the resultant vector will be the sum of the southward components.
4. Use the Pythagorean theorem to find the magnitude of the resultant vector. The magnitude of the resultant vector represents the speed of the ship relative to the ground.
5. Finally, convert the magnitude of the resultant vector from the units used in the calculations (e.g., km/hr) to the desired units, if necessary.
Following these steps, you should be able to find the speed of the ship relative to the ground.