Animated Solution for Physics - Kinematics: Consider two steamers A and B on a calm sea. Steamer A is moving towards the north with a constant speed $v_A=30\text{ km/h}$ and steamer B towards the south with a constant speed $v_B=10\text{ km/h}$. If smoke ejected by steamer A spreads in a straight line from the steamer towards the west and smoke ejected by steamer B spreads in another straight line from the steamer towards the north-west, determine magnitude and direction of the wind velocity.

The problem of the two steamers and their smoke trails is a classic, elegant puzzle in kinematics. It beautifully bridges the gap between abstract vector algebra and a highly visual, physical phenomenon. When you stand on a moving boat and watch the smoke billow from its chimney, you are not seeing the true velocity of the wind. Instead, you are witnessing a profound illusion created by your own motion. The smoke reveals the wind's velocity relative to you.

Let's embark on a journey to decode this illusion and uncover the true speed and direction of the wind sweeping across our calm sea.

Imagine you are the captain of Steamer A, cruising steadily towards the North at $30\text{ km/h}$. You look up at the smokestack and see the smoke trailing perfectly towards the West. Your intuition might scream, "Ah, the wind is blowing West!" But wait. If you were standing still on the calm sea, the smoke would indeed follow the true wind. However, because you are slicing through the air towards the North, you are creating an "apparent" wind blowing towards the South. The smoke you see is being pushed by a combination of the true wind and this apparent wind.

In physics, we formalize this using the concept of relative velocity. The velocity of the smoke as seen from the steamer is exactly the velocity of the wind relative to the steamer. Mathematically, this is expressed as: Where ${\vec{v}}_{wA}$ is the relative velocity of the wind with respect to Steamer A, ${\vec{v}}_w$ is the absolute velocity of the wind, and ${\vec{v}}_A$ is the absolute velocity of Steamer A.

To solve this elegantly, we must anchor our physical situation to a rigid mathematical coordinate system. Let's define the East direction as the positive x-axis ($+\hat{i}$) and the North direction as the positive y-axis ($+\hat{j}$).

Given this framework, we can write down the velocity vectors for our two steamers: - Steamer A is moving North at $30\text{ km/h}$: ${\vec{v}}_A=30\hat{j}$ - Steamer B is moving South at $10\text{ km/h}$: ${\vec{v}}_B=-10\hat{j}$

The true wind velocity is the mystery we need to solve. Let's represent it with unknown components:

Our goal is to find the exact values of $v_x$ and $v_y$.

Let's apply our relative velocity master equation to Steamer A. Grouping the components together, we get:

Now, we look at the crucial clue provided in the problem: the smoke from Steamer A spreads in a straight line towards the West. In our coordinate system, West corresponds to the negative x-direction ($-\hat{i}$).

If a vector points purely West, it cannot have any North or South component. This means the y-component of our relative velocity vector ${\vec{v}}_{wA}$ must be exactly zero!

Just like that, by analyzing the lack of North-South drift in Steamer A's smoke, we have uncovered the North-South component of the true wind. The wind is blowing Northwards at $30\text{ km/h}$ (in addition to whatever its East-West motion is).

Now, let's shift our perspective to Steamer B. We already know the y-component of the wind, so we can update our wind vector:

Applying the relative velocity equation for Steamer B: Notice the double negative! Because Steamer B is moving South, subtracting its velocity effectively adds a Northward component to the relative wind.

The problem states that the smoke from Steamer B spreads towards the North-West. The North-West direction is a perfect diagonal, exactly bisecting North and West. For a vector to point North-West, its Westward component and its Northward component must be perfectly balanced in magnitude.

Since the Northward (y) component is $40\text{ km/h}$, the Westward (x) component must also have a magnitude of $40\text{ km/h}$. Because it's pointing West, the value must be negative:

We have successfully decoded both clues and found the components of the true wind velocity:

To find the magnitude (speed) of the wind, we apply the Pythagorean theorem to its components:

Finally, we need to determine the direction. We can find the angle $\theta$ the wind makes with the North direction (the y-axis).

Since the x-component is negative (West) and the y-component is positive (North), the wind is blowing at an angle of approximately $53^{\circ }$ West of North.

This problem is a fantastic demonstration of how independent observations from different moving reference frames can be synthesized to reconstruct the absolute truth of a physical system. The next time you see smoke trailing from a moving vehicle, remember: you are looking at a vector subtraction problem playing out in the real world!