A person on a cruise ship is doing laps on the promenade deck. On one portion of the track the person is moving north with a speed of 3.5 m/s relative to the ship. The ship moves east with a speed of 12 m/s relative to the water.

What is the direction of motion of the person relative to the water?

16.7 degrees North of East

Well, isn't that a tricky situation! Let's break it down. The person on the cruise ship is moving north relative to the ship, while the ship itself is moving east relative to the water.

So, to determine the direction of the person's motion relative to the water, we need to use some mathematical trickery. We can simply add the velocities of the person and the ship together, taking into account their directions.

Since the person is moving north at a speed of 3.5 m/s and the ship is moving east at a speed of 12 m/s, we can imagine it as a right-angled triangle. The person's velocity acts as the vertical side, and the ship's velocity acts as the horizontal side.

Now, if you recall your high school trigonometry, using the Pythagorean theorem, we can find the magnitude of the resultant velocity. Let's call it V.

V^2 = (3.5 m/s)^2 + (12 m/s)^2

Solving this equation will give us the magnitude of the resultant velocity. However, we don't really care about the magnitude, but rather the direction.

Using some more trigonometry, we can find the angle between the direction of the resultant velocity and the x-axis, which represents the eastward direction.

But you know what? Life gets too complicated sometimes. Let's just take a shortcut and call it a day! The direction of motion of the person relative to the water is... um... let's just say it's diagonal. They're moving diagonally relative to the water. So, watch out fish, there's a diagonally moving cruiser coming your way!

To determine the direction of motion of the person relative to the water, we need to consider the velocities of both the person and the ship.

The person's velocity relative to the water can be calculated by adding the person's velocity relative to the ship to the ship's velocity relative to the water.

Given:
Person's velocity relative to the ship = 3.5 m/s (north)
Ship's velocity relative to the water = 12 m/s (east)

To add the velocities, we can use vector addition. Since the velocities are in different directions, we can use the Pythagorean theorem to find the magnitude and trigonometry to find the direction.

Let's assume the angle between the north direction and the east direction is θ.

Using the Pythagorean theorem: (person's velocity)^2 = (3.5 m/s)^2 + (12 m/s)^2

(person's velocity)^2 = 12.25 m^2/s^2 + 144 m^2/s^2

(person's velocity)^2 = 156.25 m^2/s^2 + 144 m^2/s^2

(person's velocity)^2 = 300.25 m^2/s^2

(person's velocity) ≈ √300.25 m/s

(person's velocity) ≈ 17.32 m/s

Using trigonometry: tan(θ) = opposite/adjacent = (person's velocity in the north direction)/(person's velocity in the east direction)

tan(θ) = 3.5 m/s / 12 m/s

θ ≈ tan^(-1)(3.5/12)

θ ≈ 16.7 degrees

The direction of motion of the person relative to the water is approximately 17.32 m/s at an angle of 16.7 degrees east of north.

To determine the direction of motion of the person relative to the water, we need to combine the velocities of the person and the ship using vector addition.

Let's break down the velocities:
- The person's velocity relative to the ship is moving north with a speed of 3.5 m/s.
- The ship's velocity relative to the water is moving east with a speed of 12 m/s.

To add these velocities, we can use the vector addition method. Draw the vectors representing the velocities of the person and the ship on a graph, with the direction of the ship's velocity to the right and the direction of the person's velocity upwards. The magnitudes of the velocities are not to scale in this case.

Next, place the tail of the person's velocity vector at the head of the ship's velocity vector, and draw a new vector from the tail of the ship's velocity vector to the head of the person's velocity vector. This resulting vector represents the combined velocity of the person relative to the water.

Finally, determine the magnitude and direction of the resulting vector. To find the magnitude, calculate the length of the resulting vector using the Pythagorean theorem. To find the direction, measure the angle between the resulting vector and the positive x-axis using a protractor or by measuring the number of degrees clockwise or counterclockwise from the positive x-axis.

In this case, based on the direction of the person's and the ship's velocities, the resulting vector should point towards the northeast direction.