A man is running on a straight road that makes \(30^{\circ}\) with the train track. The man is running in the direction on the road that is away from the track at a speed of \(12 \mathrm{m} / \mathrm{s}\). 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?

In physics, when an object moves in a particular direction, we can break its motion down into velocity components using the coordinate axes. Understanding these components is crucial when dealing with motion that is not along a single axis. The concept of velocity components is like splitting a force into easier-to-manage parts for analysis.

For example, let's consider an object moving at a certain angle. If we want to find out how fast the object is moving along the horizontal and vertical directions separately, we use trigonometric functions to calculate the so-called 'velocity components.' These are often referred to as the x (horizontal) and y (vertical) components of the velocity vector.

Using trigonometry, for an object moving at an angle \(\theta\), the horizontal component is the velocity multiplied by the cosine of that angle, \(\text{v}_x = v \cos(\theta)\), and the vertical component is the velocity multiplied by the sine of the angle, \(\text{v}_y = v \sin(\theta)\). This breakdown simplifies the problem significantly and provides the groundwork needed to understand and analyze motion in two dimensions.

Trigonometry is not just a topic covered in math class; it's an essential tool in physics, particularly when studying motion. It helps us understand the relationship between angles and sides of triangles, which is incredibly useful when dissecting vector quantities like velocity and force into components, as we often encounter scenarios where motion or forces are not aligned with standard coordinate axes.

In the context of our exercise, we apply trigonometric functions, sine and cosine, to split the man's velocity into horizontal and vertical components relative to the train tracks. It's important to grasp these trigonometric concepts, as misuse or misunderstanding can lead to incorrect calculation of velocity components and, subsequently, the erroneous analysis of motion.

Vector subtraction might sound complicated, but it's a way to figure out the difference in direction and magnitude between two vectors. Imagine vectors as arrows pointing in a specific direction; vector subtraction is like changing the direction one vector is pointing relative to another.

For our problem, we calculate the relative velocity, which involves finding the difference between the velocity vectors of the man and the train. The ‘subtraction’ here means we're considering the direction of the train's movement as the baseline and then determining how much faster or slower, and in which direction, the man is moving relative to the train.

This concept is vital for understanding many physical phenomena, such as relative motion, where the observed motion depends on the observer's frame of reference, which in this case is the passenger in the train.

Problems involving motion in two dimensions can be challenging, as they require an understanding of both the horizontal and vertical components of motion. This is often encountered in real-world scenarios like projectile motion, objects moving on inclined planes, or, as in our case, a man running at an angle to the direction of a moving train.

In two-dimensional motion, the object's path can be described by two independent motions that can be analyzed separately. These motions are typically along the x and y axes, or horizontal and vertical directions. We use principles such as vector addition and subtraction to combine or compare these individual motions. It's essential to handle each direction independently but also understand how they combine to affect the overall motion.

Effective problem-solving in two-dimensional motion often employs a mix of vector addition or subtraction, along with trigonometric functions to resolve the components along the axes, as demonstrated in our example.