Problem 42 - College Physics

Question

You're standing outside on a windless day when raindrops begin to fall straight down. You run for shelter at a speed of \(5.0 \mathrm{m} / \mathrm{s},\) and you notice while you're running that the raindrops appear to be falling at an angle of about \(30^{\circ}\) from the vertical. What's the vertical speed of the raindrops?

Step-by-Step Solution

Answer

The vertical speed of the raindrops is approximately 8.66 m/s.

Step 1: Understanding the Problem

You are moving at a speed of 5.0 m/s and observe the raindrops to be falling at an angle of 30° from the vertical. We need to find the actual vertical speed of the raindrops when they are falling straight down.

Step 2: Identifying Components of Raindrop Velocity

When observed at an angle, the raindrop velocity has two components: vertical and horizontal. The horizontal component equals your running speed (5.0 m/s), because it's what makes the rain seem to angle when you're moving.

Step 3: Using Trigonometric Principles

The tangent of the observed angle relates the horizontal component of velocity (your speed) and the vertical component (actual vertical speed of the raindrops). Given angle \( \theta = 30^{\circ} \), you can write:\[\tan(30^{\circ}) = \frac{v_{\text{horizontal}}}{v_{\text{vertical}}} = \frac{5.0}{v_{\text{vertical}}}\]

Step 4: Solving for the Vertical Speed

Rearrange the tangent equation to solve for the vertical speed \(v_{\text{vertical}}\):\[v_{\text{vertical}} = \frac{5.0}{\tan(30^{\circ})}\]Substitute the value \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \approx 0.577 \):\[v_{\text{vertical}} = \frac{5.0}{0.577} \approx 8.66 \text{ m/s}\]

Step 5: Conclusion

The vertical speed of the raindrops, when falling straight down, is approximately 8.66 m/s.

Key Concepts

Trigonometry in Physics

Trigonometry is a pivotal tool in physics, particularly when it comes to analyzing motion in different directions.
It helps us break down components like velocity into more manageable parts. For instance, when dealing with projectile motion, we often need to understand both horizontal and vertical components separately.
In this exercise, raindrops falling at an angle represent a classic example of how trigonometry is used in physics.

• We use trigonometric functions, such as \( \tan \), to relate angles to the ratios of sides in a right triangle.
• The formula \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) lets us calculate the vertical speed component when the horizontal speed and angle are known.

By breaking a complex problem into simpler parts, trigonometry allows us to solve for unknown physical quantities, making it indispensable in physics.

Relative Velocity

Relative velocity refers to the velocity of an object from the perspective of another object.
It is important when analyzing how motion appears from different reference points.
In this scenario, the observer (you) runs at a speed of 5.0 m/s, which becomes the horizontal component of the observed raindrop velocity.
When raindrops appear to fall at an angle, it's because your horizontal motion alters the perspective.
This perceived motion and the actual motion of the raindrops are different due to relative velocity.

• With the observer moving, the relative horizontal velocity of the raindrops matches the observer's speed.
• The relative vertical velocity needs to be calculated, which involves separating it from the horizontal motion using trigonometric principles.

Understanding relative velocity helps explain why objects appear to move differently to different observers, fully integrating both physics and trigonometry.

Angle of Motion

The angle of motion is not just a measure of direction; it's essential for understanding the relationship between different components of motion.
In this exercise, the raindrops fall straight down, but the perceived 30° angle emerges due to your movement.
This perception shift is significant because it links directly to the concept of projectile motion.

• The angle dictates how much of the total velocity is distributed vertically and horizontally.
• A 30° angle implies a specific ratio of vertical to horizontal movement, calculable through trigonometric identities.

By knowing the angle and one component of motion (here, the horizontal speed), we can accurately compute the other component, deepening our understanding of motion in physics.