A man is running on a straight road perpendicular to a train track and away from the track at a speed of \(12 \mathrm{m} /\) s. The train is moving with a speed of \(30 \mathrm{m} / \mathrm{s}\) with respect to the track. What is the speed of the man with respect to a passenger sitting at rest in the train?

Understanding relative motion is crucial when analyzing how different objects move with respect to each other. It's about discerning how the movement of one object looks from the standpoint of another moving object.

In physics, the concept of relative motion tells us that the velocity of an object is always measured with respect to another object or a frame of reference. This can lead to differing observations depending on the observer's state of motion. For instance, if you're on a train passing by trees, the trees seem to move backward when, in reality, they are stationary, and it's the train that's moving. This phenomenon is a simple demonstration of relative motion.

When solving relative velocity problems, like the textbook example, it's essential to determine the speed of one object as observed from another moving object. In this case, we find the speed of the man as observed by a passenger on the train. By applying relative motion principles, we understand that both the direction and the magnitude of the two velocities need to be considered to find the relative speed between the man and the train.

The application of the Pythagorean theorem extends far beyond geometry into various fields, including physics—particularly in relative velocity problems involving perpendicular motion. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This principle is handily utilized when two objects move at right angles to each other, as in our example with the man and the train. Their paths form the perpendicular sides of an imaginary right-angled triangle, with their relative velocity vector forming the hypotenuse.

To calculate relative velocity in such a scenario, you square the speed of each object, sum these values, and then take the square root of the result. Following the textbook solution, we have effectively applied the Pythagorean theorem to determine the man's speed with respect to the train, which is found to be approximately 32.31 meters per second.

Frames of reference are integral to solving problems involving motion. They provide a perspective to measure and observe motion. A frame of reference can be thought of as a set of coordinate axes which can be stationary or moving.

These frames are essential because, as seen in relative motion, the description and the numerical value of velocity depend on the observer's frame of reference. There are two types of frames of reference: inertial, where Newton's first law holds, and non-inertial, which is subject to acceleration.

In our textbook problem, the frame of reference for the passenger is the train, which we assume to be moving at a constant velocity, thereby acting as an inertial frame for the purposes of the problem. By adopting the perspective of the passenger, we are able to use the train as our frame of reference for calculating the relative velocity of the man, exemplifying how choosing your frame of reference can impact the outcome and understanding of a physics problem.