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Chapter 3: Problem 42

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?

Short Answer

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

Step by step solution

01

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掳 from the vertical. We need to find the actual vertical speed of the raindrops when they are falling straight down.
02

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.
03

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:\[\tan(30^{\circ}) = \frac{v_{\text{horizontal}}}{v_{\text{vertical}}} = \frac{5.0}{v_{\text{vertical}}}\]
04

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 \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \approx 0.577 \):\[v_{\text{vertical}} = \frac{5.0}{0.577} \approx 8.66 \text{ m/s}\]
05

Conclusion

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳 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掳 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.

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Most popular questions from this chapter

\(\bullet\) Dynamite! A demolition crew uses dynamite to blow an old building apart. Debris from the explosion flies off in all directions and is later found at distances as far as 50 \(\mathrm{m}\) from the explosion. Estimate the maximum speed at which debris was blown outward by the explosion. Describe any assumptions that you make.

\(\bullet\) Bird migration. Canadian geese migrate essentially along a north- south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 \(\mathrm{km} / \mathrm{h}\) . If one such bird is flying at 100 \(\mathrm{km} / \mathrm{h}\) relative to the air, but there is a 40 \(\mathrm{km} / \mathrm{h}\) wind blowing from west to east, (a) at what angle relative to the north-south direction should this bird head so that it will be traveling directly southward relative to the ground? (b) How long will it take the bird to cover a ground distance of 500 \(\mathrm{km}\) from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth's magnetic field to fix the north-south direction.)

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very high 3. 0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 \(\mathrm{cm}\) from the axis of spin. What is the radial acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} s\) s of the endolymph fluid?

\bulletA test rocket is launched by accelerating it along a \(200.0-\mathrm{m}\) incline at 1.25 \(\mathrm{m} / \mathrm{s}^{2}\) starting from rest at point \(A\) (Figure \(3.40 . )\) The incline rises at \(35.0^{\circ}\) above the horizontal,and at the instant the rocket leaves it, its engines turn off and it is subject only to gravity (air resistance can be ignored). Find (a) the maximum height above the ground that the rocket reaches, and (b) the greatest horizontal range of the rocket beyond point \(A .\)

Firemen are shooting a stream of water at a burning building. A high-pressure hose shoots out the water with a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) as it leaves the hose nozzle. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00 s to reach a building 45.0 m away. You can ignore air resistance; assume that the end of the hose is at ground level.(a) Find the angle of elevation of the hose. (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast visit moving just before it hits the building?
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