Question

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 \(3.8 \mathrm{~m} / \mathrm{s}\) relative to the ship. The ship moves east with a speed of \(12 \mathrm{~m} / \mathrm{s}\) relative to the water. What is the direction of motion of the person relative to the water?

Short answer

The person's direction relative to the water is 17.6° north of east.

Step-by-step solution

Understand the Velocity Components

We're given the northward velocity of the person relative to the ship ( 3.8 m/s) and the eastward velocity of the ship relative to the water (12 m/s). The velocities form a right triangle.

Calculate the Resultant Velocity Magnitude

To find the magnitude of the velocity of the person relative to the water, use the Pythagorean theorem:\[ v = \sqrt{(3.8)^2 + (12)^2} \]

Solve for Magnitude

Calculate the magnitude:\[ v = \sqrt{3.8^2 + 12^2} = \sqrt{14.44 + 144} = \sqrt{158.44} \approx 12.59 \text{ m/s} \]

Determine Direction Angle

Use trigonometry to find the angle of the person's velocity relative to the water. Use the tangent function:\[ \theta = \tan^{-1}\left(\frac{3.8}{12}\right) \]

Calculate the Angle

Calculate \( \theta \):\[ \theta \approx \tan^{-1}(0.3167) \approx 17.6^\circ \]

Interpret the Result

The direction of motion of the person relative to the water is approximately 17.6° north of east.

Key concepts

Vector Addition

When dealing with relative velocity problems, **vector addition** is a critical tool. Imagine you have different motion pathways represented by vectors. For example, in our problem, the person moves north relative to the ship while the ship moves east relative to the water. These motions are their own vectors. By adding these vectors, you'll determine the overall motion relative to the water. To perform vector addition, draw each vector representing a different motion. Then, using the head-to-tail method, place the tail of one vector at the head of another. The resultant vector, drawn from the tail of the first vector to the head of the last vector, represents the overall velocity.
This visualization helps in understanding the compound effect of multiple movements and eases the process of subsequent calculations.

Whenever you have such a problem, remember:

• Identify each motion as a vector.
• Use the head-to-tail method to combine them.
• The resulting vector gives the direction and magnitude of combined motion paths.

This approach simplifies complex relationships between combined paths and opens the door to using mathematical tools like trigonometry.

Trigonometry

In the study of vectors and motion, **trigonometry** provides the framework to calculate angles and distances accurately. Once you combine vectors using vector addition, the next step often involves determining the direction or angle of the resultant vector. This is where trigonometry comes into play.
With the example exercise, the person’s northward movement and the ship’s eastward movement form a right triangle. Here, we can use trigonometric ratios such as tangent, sine, or cosine to find the relationship between the angles and the sides of the triangle.

The tangent function is especially handy when you have a right triangle and need to find the angle of the resultant vector. It is defined as the ratio between the opposite side and the adjacent side in a right triangle:

• tan(𝜃) = opposite/adjacent

In our example, this means using the speeds to find the angle:

• tan(𝜃) = 3.8 m/s (north) / 12 m/s (east)

Use the inverse tangent function (tan^-1) to compute the angle 𝜃, which gives the direction relative to the water.
This approach is fundamental in physics and engineering, where precise direction and magnitude calculations are often needed.

Pythagorean Theorem

The **Pythagorean theorem** is a simple yet powerful tool for finding the magnitude of a resultant vector when dealing with perpendicular vectors. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In velocity problems where the direction of movement is perpendicular, this theorem becomes particularly useful. For example, if you know the northward and eastward components of a person's velocity, as in our exercise, you can apply the theorem to find the overall velocity relative to a stationary observer, like the water:

• If you have sides of 3.8 m/s and 12 m/s, the formula becomes:
• v = \( \sqrt{(3.8)^2 + (12)^2} \)

Calculate this to find v, the magnitude of the person's velocity relative to the water.
This theorem simplifies the process of finding resultant velocities in perpendicular motion problems efficiently. Its applications extend beyond physics, into everyday scenarios where measures need totaling along perpendicular axes.