Question

A cruise ship moves southward in still water at a speed of \(20.0 \mathrm{~km} / \mathrm{h}\) while a passenger on the deck of the ship walks toward the east at a speed of \(5.0 \mathrm{~km} / \mathrm{h}\). The passenger's velocity with respect to Earth is a) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) east of south. b) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) south of east. c) \(25.0 \mathrm{~km} / \mathrm{h}\), south. d) \(25.0 \mathrm{~km} / \mathrm{h},\) east. e) \(20.6 \mathrm{~km} / \mathrm{h}\), south.

Short answer

Based on the given information and step by step solution, complete the short answer below. The speed of the passenger with respect to Earth is approximately 20.6 km/h, and the direction of their motion is 14.04 degrees east of south.

Step-by-step solution

Calculate the velocity of the passenger with respect to Earth

To calculate the passenger's velocity with respect to Earth, we need to add the velocity vector of the cruise ship (southward) and the velocity vector of the passenger on the ship (eastward). We can represent the velocity vectors as `<20, 0>` (south) for the cruise ship and `<0, 5>` (east) for the passenger. Now, we sum the two vectors to get the resulting velocity vector `<20, 5>`, which represents the passenger's velocity with respect to Earth.

Calculate the magnitude of the passenger's velocity with respect to Earth

Now that we have the passenger's velocity vector with respect to Earth `<20, 5>`, we calculate the magnitude of the velocity vector using the Pythagorean theorem: \(v = \sqrt{20^2 + 5^2} = \sqrt{400 + 25} = \sqrt{425} \approx 20.6 \mathrm{~km} / \mathrm{h}\) This means that the magnitude of the passenger's velocity with respect to Earth is about \(20.6 \mathrm{~km} / \mathrm{h}\).

Calculate the direction angle of the passenger's velocity with respect to Earth

To find the direction angle of the passenger's velocity with respect to Earth, we can use trigonometry. The angle can be found by using the arctangent function: \(\theta = \arctan(\frac{5}{20})\) \(\theta \approx 14.04^{\circ}\) This angle is measured from the southward direction, towards the east (counterclockwise). So the direction is "east of south".

Identify the correct answer

Now that we have calculated the magnitude of the passenger's velocity with respect to Earth (\(20.6 \mathrm{~km} / \mathrm{h}\)) and the direction angle (\(14.04^{\circ}\) east of south), we can identify the correct choice among the given options: a) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) east of south. This choice matches the calculated values perfectly, so the correct answer is: a) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) east of south.

Key concepts

Vector Addition

When we talk about objects moving, we're looking at their velocities, which are actually vectors. This means they have both magnitude (how fast something is moving) and direction (where it's heading).

Imagine you're on a train moving forward and you start walking to the snack car. To someone outside, your movement is a combination of the train's movement and your own. That's vector addition! We simply put together the train's velocity vector with your walking velocity vector to find out your actual velocity compared to the ground.

With the cruise ship and the passenger, we added the ship's southward velocity vector to the passenger's eastward walking vector. When we combine vectors, we add the corresponding components. For the ship and passenger, their combined vector is a diagonal one representing their true course and speed over the ground.

Pythagorean Theorem

Important in many fields, the Pythagorean theorem is a cornerstone of geometry and finds its place in physics, especially when dealing with vectors. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The formula is written as: \( c^2 = a^2 + b^2 \), where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the triangle's other two sides. This theorem allows us to calculate the magnitude of a velocity vector by treating its components as sides of a right triangle. In the cruise ship example, the Pythagorean theorem helped us find the resulting speed of the passenger relative to the Earth.

Trigonometry in Physics

Trigonometry plays a pivotal role in physics, helping us describe and analyze the components of vectors, including velocity and force. When you have one vector pointing one way and another at a perpendicular angle, trigonometry lets us find the direction of the resultant vector.

Functions like sine, cosine, and tangent are frequently used. For instance, to find out the angle at which the passenger's resulting velocity points, we've used the arctangent function. This function, often written as \( \arctan \), gives us the angle whose tangent is the division of the opposite side by the adjacent side of a right-angled triangle. In the ship example, the calculation told us the angle the passenger's path makes with the southward direction.

Velocity Magnitude and Direction

In physics, the complete description of velocity requires two things: how fast an object is moving (magnitude) and which way it's going (direction). Magnitude is simply the speed component and is typically measured in units like meters per second (m/s) or kilometers per hour (km/h).

Direction, meanwhile, can be described using angles or cardinal directions (like north or east). In our ship example, the magnitude is approximately 20.6 km/h. For direction, we used trigonometry to find the bearing, which turned out to be 14.04 degrees east of south. The combination of magnitude and direction gives us a complete picture of the passenger's actual velocity relative to Earth.