Question

A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of his velocity relative to the surface of the water?

Short answer

The man's velocity relative to the water is approximately 20.22 mph at an angle of 8.53° west of south.

Step-by-step solution

Identify Motion Components

The man on the ship has two components of motion: the ship's southward speed, and his own westward speed on the deck. The ship moves south at 20 mph while the man walks west at 3 mph relative to the ship.

Set Up Velocity Vectors

Represent the southward speed of 20 mph as the vector \( \vec{v}_s = (0, -20) \) (in miles per hour) and the westward speed of 3 mph as \( \vec{v}_m = (-3, 0) \). The velocities are perpendicular.

Calculate Resultant Velocity

Use vector addition to find the man's resultant velocity relative to the water: \( \vec{v}_r = \vec{v}_s + \vec{v}_m = (-3, 0) + (0, -20) = (-3, -20) \).

Determine Magnitude of Velocity

Calculate the magnitude of the resultant velocity vector using the Pythagorean theorem: \( \|\vec{v}_r\| = \sqrt{(-3)^2 + (-20)^2} = \sqrt{9 + 400} = \sqrt{409} \approx 20.22 \) mph.

Find Direction of Velocity

Find the direction as an angle \( \theta \) relative to the southward direction using the tangent function: \( \tan(\theta) = \frac{3}{20} \). Compute \( \theta = \tan^{-1}\left(\frac{3}{20}\right) \approx 8.53^\circ \), which is west of south.

Key concepts

Resultant Velocity

When dealing with motion, especially in multiple directions, it is important to determine the resultant velocity. The resultant velocity tells us the overall speed and direction of an object. In the scenario provided, a man on a ship has two velocity components: the ship moving south at 20 mph and the man walking west at 3 mph.

To find the resultant velocity, we combine these two separate motions. The ship's southward motion and the man's westward motion create a diagonal path relative to the surface of the water. This diagonal path is called the resultant of the two velocities. It represents how an observer on land would see the man moving.

This concept is crucial when objects move in different directions simultaneously, as it provides a complete picture of the movement.

Vector Addition

Vector addition is a mathematical operation to find the resultant of two or more vectors. In this case, we have two velocity vectors: the ship's southward speed and the man's westward speed.

Vectors have both magnitude (strength of motion) and direction. To add vectors correctly, it's important to consider their direction. We use their horizontal and vertical components to find the resultant vector.

For this exercise, the ship's velocity vector is represented as \( \vec{v}_s = (0, -20) \) (southward direction) and the man's walking velocity as \( \vec{v}_m = (-3, 0) \) (westward direction). By adding these vectors component-wise, we find the resultant velocity vector: \( \vec{v}_r = (-3, -20) \). This method ensures that both the speed and direction are accurately accounted for.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that helps us calculate the length of the hypotenuse of a right-angled triangle. In vector analysis, it is used to determine the magnitude of a resultant vector.

In this problem, the two velocity components form a right-angled triangle, with the resultant velocity as the hypotenuse. The theorem states that the square of the hypotenuse (resultant velocity) is equal to the sum of the squares of the other two sides (individual velocity components).

Mathematically, it is expressed as:
\[\|\vec{v}_r\| = \sqrt{(-3)^2 + (-20)^2} = \sqrt{409} \approx 20.22 \text{ mph}\]This tells us the speed with which the man moves relative to the water.

Trigonometric Functions

Trigonometric functions, such as tangent, sine, and cosine, are used to find angles in triangle-related problems. In vector calculations, these functions help determine the direction of the resultant vector.

In this example, the direction of the man's velocity relative to due south is found using the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side (westward speed) to the adjacent side (southward speed).

The formula is: \[\tan(\theta) = \frac{3}{20}\]By finding the inverse tangent, we get the angle \( \theta \):\[\theta = \tan^{-1}\left(\frac{3}{20}\right) \approx 8.53^\circ\] This angle represents how far west of south the man's velocity is. Understanding these trigonometric principles helps in accurately describing directions in real-world motion.