At the entrance channel of a harbor, the tidal current has a velocity of 4.22 km/hr in a direction 17.4o south of east. Suppose a ship caught in this current has a speed of 17.5 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?

Vw = 4.22 km/h[17.4o] S. of E. = 4.22km/h[ 355.8o] CCW.

Vs = 17.5km/h[90o] CCW.

Vr = 4.22[17.4o ]+ 17. 5[90o],
X = 4.22*Cos17.4 + 17.5*Cos90 = 4.03 km.
Y = 4.22*sin17.4 + 17.5*sin90 = 18.8 km.
Vr = 4.03 + 18.8i = 19.2 km[77.9o] CCW. = Resultant velocity of ship.

To find the speed of the ship relative to the ground, we need to consider both the velocity of the tidal current and the velocity of the ship relative to the water.

First, let's break down the velocity of the tidal current into its components. The velocity of the current has a magnitude of 4.22 km/hr and is directed at an angle of 17.4 degrees south of east. We can find the eastward and southward components of the current using trigonometry.

The eastward component of the tidal current velocity is given by:
Eastward component = magnitude of current velocity * cos(angle)
= 4.22 km/hr * cos(17.4 degrees)

The southward component of the tidal current velocity is given by:
Southward component = magnitude of current velocity * sin(angle)
= 4.22 km/hr * sin(17.4 degrees)

Now, let's consider the ship's velocity relative to the water. The ship has a speed of 17.5 km/hr and is aimed north. Since the ship is aimed north and the water is flowing in a different direction, we need to account for the angle between the ship's heading and the current direction.

Since we know the ship's heading is north, the angle between the ship's heading and the eastward direction is 90 degrees. Therefore, the northward component of the ship's velocity relative to the water is given by:
Northward component = speed of ship relative to water * cos(90 degrees)
= 17.5 km/hr * cos(90 degrees)

Finally, to find the ship's speed relative to the ground, we need to combine the velocities of the tidal current and the ship relative to the water. We sum the eastward, southward, and northward components to get the resultant velocity.

The speed of the ship relative to the ground is given by:
Resultant velocity = sqrt((Eastward component + Northward component)^2 + Southward component^2)

Substituting the values we calculated for the components, we can find the speed of the ship relative to the ground.