Final answer:
To find the resultant velocity of the ship, we decompose the ocean current's velocity into its northward and eastward components, then sum these with the ship's velocity. After calculating the total northward and eastward velocities, we determine the magnitude and direction of the resultant velocity using vector addition and trigonometric functions.
Step-by-step explanation:
To calculate the resultant velocity of the ship, we will use vector addition. The ship's velocity is due north at 8.5 m/s, and the ocean current's velocity is 1.50 m/s at 40.0° north of east.
First, we need to decompose the ocean current's velocity into its northward and eastward components:
- Northward component (V_north) = 1.50 m/s × cos(40.0°)
- Eastward component (V_east) = 1.50 m/s × sin(40.0°)
Next, we add the components of the ocean current to the ship's velocity:
- Total northward velocity (V_Tnorth) = 8.5 m/s + V_north
- Total eastward velocity (V_Teast) = V_east
Now, we calculate the magnitude of the resultant velocity (V_R):
V_R = √(V_Tnorth² + V_Teast²)
Finally, we find the direction θ relative to the north (eastward direction is taken as positive):
θ = tan⁻¹(V_Teast / V_Tnorth)
By performing these calculations, we obtain the speed and direction of the ship relative to the Earth. The correct answer from the options provided should be determined based on the calculated values of V_R and θ.