Problem 43 - College Physics

Question

\(\bullet\) 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.)

Step-by-Step Solution

Answer

(a) 23.58° west of south, (b) 5.43 hours.

Step 1: Identify the Problem

The question provides the bird's velocity relative to the air as 100 km/h toward the south and a wind blowing at 40 km/h from west to east. We need to find the angle the bird should maintain to move south relative to the ground.

Step 2: Resolve the Velocity Vectors

The bird’s velocity relative to the ground is the vector sum of its velocity relative to the air and the wind velocity. Denote the bird's required southward velocity as V and the angle it makes with the south-north direction as \( \theta \).

Step 3: Set Up Vector Components

The bird’s velocity components are given by:\[ V_{north} = 100 \cos(\theta) \]\[ V_{east} = 100 \sin(\theta) \] The wind's velocity component is 40 km/h eastward.

Step 4: Ensure No Effective Eastward Component

To ensure the bird travels directly south, the eastward components must cancel:\[ 100 \sin(\theta) = 40 \]

Step 5: Solve for the Angle \(\theta\)

Re-arrange to solve for \( \theta \):\[ \sin(\theta) = \frac{40}{100} = 0.4 \]\[ \theta = \arcsin(0.4) \approx 23.58^\circ \] This means the bird should head 23.58° west of south.

Step 6: Calculate the Effective Southward Speed

Use the cosine of \(\theta\) to find the southward ground speed:\[ V_{south} = 100 \cos(23.58^\circ) \approx 100 \times 0.92 = 92 \text{ km/h} \]

Step 7: Determine the Time for Journey

To find the time taken to travel 500 km south:\[ \text{Time} = \frac{500 \text{ km}}{92 \text{ km/h}} \approx 5.43 \text{ hours} \]

Key Concepts

Bird Migration and Orientation

Bird migration is an amazing natural phenomenon, where many species travel long distances to reach their breeding or feeding grounds. Canadian geese, for example, can travel well over a thousand kilometers north or south, depending on the season. These geese fly at impressive speeds, sometimes reaching up to 100 km/h when conditions are favorable. Despite facing varied challenges such as weather conditions and natural obstacles, birds utilize the Earth's magnetic field to maintain their directionality. This ability allows them to travel accurately even when visual cues like stars or the sun are not present. Understanding migration patterns is essential for recognizing the impact of environmental changes on bird populations worldwide.

Birds use a variety of sensory information:

• Earth's magnetic field for navigation.
• Sun and stars as directional guides.
• Mental maps and learned pathways.

These tools help them to adjust their course and ensure successful migration.

Understanding Relative Velocity in Bird Flight

In the context of bird flight, relative velocity is crucial for determining the actual path a bird takes in relation to the ground. Relative velocity considers the bird's velocity in the air as well as external factors such as the wind. Suppose a bird is flying at a constant speed of 100 km/h relative to the air, but a wind blowing at 40 km/h is coming from the west. In that case, relative velocity helps us compute the bird's trajectory concerning the ground.

The concept of relative velocity in physics is about understanding how different forces interact to affect an object's motion. In bird migration, it is vital since the bird must adjust its flight path accordingly to maintain its intended direction.

• Bird's velocity relative to air: primary speed of bird.
• Wind velocity relative to ground: environmental factor affecting trajectory.
• Ground velocity: combined effect of bird's and wind speed.

By resolving these velocity vectors, one can determine the angle the bird should fly to counter the wind and maintain a southward trajectory.

Wind Effects on Bird Trajectory

Wind significantly impacts bird migration by altering flight paths and affecting navigation. In the exercise, the 40 km/h wind blowing from west to east requires the bird to compensate by adjusting its flight angle. This adjustment helps it maintain a direct path southwards relative to the ground. Understanding how wind affects movement is vital in vector analysis, as it involves examining both the wind's speed and direction to predict the bird's necessary course adjustments.

When calculating wind effects, consider the vector components of both the bird's flight speed and the wind direction. The objective is to cancel out unwanted eastward motion:

For example:

• Bird's eastward component: 100 sin(θ)
• Wind's eastward component: 40 km/h
• To maintain a southward path: 100 sin(θ) = 40, resulting in θ ≈ 23.58° west of south

This illustrates how birds require a precise understanding of wind effects to achieve optimal navigation and energy efficiency during migration.

Applying Trigonometry in Physics for Bird Paths

Trigonometry is crucial in physics when analyzing vectors, like the velocity vectors involved in bird migration. In the exercise, trigonometry helped determine the angle at which the bird should fly to maintain a southward path despite crosswinds. Use sine and cosine functions to resolve forces into perpendicular components that can be managed separately.

In the problem, the bird's velocity against the wind is broken down into components using trigonometric functions:

• Bird's north-south component: 100 cos(θ)
• Bird's east-west component: 100 sin(θ)

The sine function aids in canceling out eastward wind effects by solving the equation 100 sin(θ) = 40, ensuring no net eastward motion.

Trigonometry in such vector problems allows for precise calculations of angles and speeds, ensuring accurate navigation despite external forces like wind. By using trigonometric relationships, you can solve for unknowns and thus determine the most efficient path for migratory birds.