A ship sets sail from Rotterdam, The Netherlands, heading due north at 5.00 m/s relative to the water. The local ocean current is 1.54 m/s in a direction 40° north of east. What is the velocity of the ship relative to the earth?
To find the velocity of the ship relative to the earth, we need to find the vector sum of the ship's velocity relative to the water and the velocity of the water relative to the earth.
First, let's represent the vectors in terms of their components. The ship's velocity relative to the water can be split into two components: the northward component and the eastward component. Since it is heading due north, the northward component is 5.00 m/s and the eastward component is 0 m/s.
Next, let's find the components of the current velocity. The angle given is 40° north of east. To find the eastward component, we can use the cosine function: cos(40°) = adjacent/hypotenuse. Since the hypotenuse is the magnitude of the current velocity (1.54 m/s), the eastward component is calculated as follows:
eastward component = cos(40°) × 1.54 m/s
Similarly, to find the northward component, we can use the sine function: sin(40°) = opposite/hypotenuse. The northward component is:
northward component = sin(40°) × 1.54 m/s
Now, add up the respective components to find the total velocity relative to the earth:
northward component (ship's velocity) + northward component (current velocity) = total northward component
eastward component (ship's velocity) + eastward component (current velocity) = total eastward component
The total velocity relative to the earth can be calculated using the Pythagorean theorem:
total velocity = square root of (total northward component^2 + total eastward component^2)
By plugging in the respective values, you can find the answer.