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 did this using the Law of Cosines and came up with 19.14 km/hr. However, this is not right. What did I do wrong?
To solve this problem, you correctly used the Law of Cosines to find the magnitude of the ship's velocity relative to the ground. However, you made a mistake in determining the angle between the ship's velocity and the tidal current.
To find the correct angle between the ship's velocity and the tidal current, you need to add the angle of the tidal current (23.2° south of east) to the angle of the ship's velocity relative to the water (which is directly north). Since north is 90° counterclockwise from east, the angle between the ship's velocity and the tidal current is:
23.2° + 90° = 113.2°
Now you can use the Law of Cosines to find the ship's velocity relative to the ground. Let's call this velocity V:
V² = (15.6 km/hr)² + (4.94 km/hr)² - (2 * 15.6 km/hr * 4.94 km/hr * cos(113.2°))
Now, calculate the correct value of V using the above equation.
To find the speed of the ship relative to the ground, we can break down the ship's velocity into its components, one parallel to the direction of the current and the other perpendicular to it.
First, let's find the components of the current velocity. The given velocity of the current is 4.94 km/hr at an angle of 23.2° south of east.
The eastward component of the current velocity can be found by multiplying the magnitude of the current velocity (4.94 km/hr) by the cosine of the angle (23.2°):
Eastward Component: 4.94 km/hr * cos(23.2°) = 4.36 km/hr (approximately)
The southward component of the current velocity can be found by multiplying the magnitude of the current velocity (4.94 km/hr) by the sine of the angle (23.2°):
Southward Component: 4.94 km/hr * sin(23.2°) = 2.09 km/hr (approximately)
Now, let's find the components of the ship's velocity relative to the water. The ship's velocity relative to the water is given as 15.6 km/hr, and because the ship is aimed north, there is no eastward component of velocity to consider. Therefore:
Northward Component: 15.6 km/hr
Now we can add the components of the current and the ship's velocity relative to the water to find the ship's velocity relative to the ground. Both the northward and southward components contribute to the resultant velocity:
Northward Component of Resultant Velocity = Northward Component of Ship's Velocity + Southward Component of Current Velocity
= 15.6 km/hr + 2.09 km/hr = 17.69 km/hr (approximately)
Eastward Component of Resultant Velocity = Eastward Component of Current Velocity
= 4.36 km/hr (approximately)
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
Resultant Velocity = √((Northward Component of Resultant Velocity)^2 + (Eastward Component of Resultant Velocity)^2)
= √((17.69 km/hr)^2 + (4.36 km/hr)^2)
= √(312.76 km^2/hr^2 + 19.04 km^2/hr^2)
= √331.8 km^2/hr^2
= 18.21 km/hr (approximately)
Therefore, the correct speed of the ship relative to the ground is approximately 18.21 km/hr, not 19.14 km/hr as you calculated using the Law of Cosines.