heading due north at 6.00 m/s relative to the water. The local ocean current is 1.53 m/s in a direction 40° north of east. What is the velocity of the ship relative to the earth?
It is the vector sum of the velocity of the ship relative to the water, and the velocity of the water with respect to the land.
V = 6.0m/s[90o] + 1.53m/s[40o]
X = 6.0*Cos90 + 1.5*Cos40 = 0 + 1.15 = 1.15 m/s.
Y = 6*sin90 + 1.5*sin40 = 6 + 0.96 = 6.96 m/s.
Tan A = Y/X = 6.96/1.15 = 6.05.
A = 80.6o.
V = Y/sin A = 6.96/sin80.6 = 7.05 m/s[80.6o]
To find the velocity of the ship relative to the Earth, you need to combine the ship's velocity due north and the ocean current's velocity.
Step 1: Break down the ship's velocity into its horizontal (x-axis) and vertical (y-axis) components:
- The ship's velocity due north is 6.00 m/s along the y-axis.
Step 2: Break down the ocean current's velocity into its horizontal and vertical components:
- The direction of the ocean current is 40° north of east.
- We can calculate the x-axis component by using the cosine function: cos(40°) = adjacent/hypotenuse;
=> x-axis component = 1.53 m/s * cos(40°).
- We can calculate the y-axis component by using the sine function: sin(40°) = opposite/hypotenuse;
=> y-axis component = 1.53 m/s * sin(40°).
Step 3: Combine the x-axis and y-axis components of the ship's velocity and the ocean current's velocity to get the resulting velocity:
- The x-axis component of the resulting velocity is the sum of the ship's x-axis velocity and the ocean current's x-axis velocity.
- The y-axis component of the resulting velocity is the sum of the ship's y-axis velocity and the ocean current's y-axis velocity.
Step 4: Calculate the magnitude and direction of the resulting velocity vector using the Pythagorean theorem and the tangent function:
- The magnitude of the resulting velocity vector is given by: magnitude = sqrt(squared_x_axis_component + squared_y_axis_component).
- The direction of the resulting velocity vector is given by: direction = atan(y_axis_component / x_axis_component).
By following these steps, you can calculate the velocity of the ship relative to the Earth.