A person is walking at a speed of 3 km/hr. He sees that the rain is falling down vertically. He increases his speed to 6 km/hr. The rain appears to meet him at an angle of 45 degrees with the vertical. Find the speed of the rain ?

To solve this problem, we need to analyze the situation using some basic principles of relative motion. The key here is to understand how the speed of the person and the rain interact to create the observed angle. Let's break it down step by step.

Understanding the Scenario

Initially, the person is walking at a speed of 3 km/hr, and he observes the rain falling vertically. When he increases his speed to 6 km/hr, he notices that the rain appears to be coming at him at a 45-degree angle from the vertical. This change in perception is crucial for determining the speed of the rain.

Relative Motion Concept

Relative motion refers to how the speed of one object appears to change when viewed from another moving object. In this case, the person is moving, and we need to consider how his movement affects his perception of the rain's speed.

Setting Up the Problem

Let’s denote:

• Vr = speed of the rain (in km/hr)
• Vp = speed of the person = 6 km/hr (after he increases his speed)

When the rain appears to meet him at a 45-degree angle, we can use trigonometry to relate the speeds. The angle of 45 degrees indicates that the horizontal component of the rain's speed (as perceived by the person) is equal to the vertical component of the rain's speed.

Breaking Down the Components

Since the rain is falling vertically, its vertical speed is simply Vr. The horizontal speed, as perceived by the person, is equal to his speed, which is 6 km/hr. At a 45-degree angle, we have:

• Horizontal component = Vp = 6 km/hr
• Vertical component = Vr

For an angle of 45 degrees, the relationship between the horizontal and vertical components can be expressed as:

Finding the Speed of the Rain

Since we know that the horizontal speed (6 km/hr) equals the vertical speed (Vr), we can set them equal to each other:

Vr = 6 km/hr

Thus, the speed of the rain is 6 km/hr. This means that the rain is falling at a speed of 6 km/hr vertically, which, when combined with the person's horizontal movement, creates the observed angle of 45 degrees.

Conclusion

In summary, by analyzing the relative motion and applying basic trigonometric principles, we determined that the speed of the rain is 6 km/hr. This example illustrates how relative speeds can affect our perception of motion in different directions.