A Bird Is Flying At $25.5 \, \text{m/s}$ In The $y$-direction. The Wind Is Blowing At $3.95 \, \text{m/s}$ In The $x$-direction.What Is The Direction Of The Velocity Of The Bird?$\begin{array}{c} \theta =

Introduction

When a bird flies in the presence of wind, its velocity is affected by the wind's direction and speed. In this scenario, we have a bird flying at a speed of $25.5 , \text{m/s}$ in the $y$-direction, while the wind is blowing at $3.95 , \text{m/s}$ in the $x$-direction. To determine the direction of the bird's velocity, we need to consider the components of its velocity and the wind's velocity.

Understanding Velocity Components

Velocity is a vector quantity, which means it has both magnitude (speed) and direction. In this case, the bird's velocity has a magnitude of $25.5 , \text{m/s}$ and is directed in the $y$-direction. The wind's velocity, on the other hand, has a magnitude of $3.95 , \text{m/s}$ and is directed in the $x$-direction.

Calculating the Resultant Velocity

To find the direction of the bird's velocity, we need to calculate the resultant velocity, which is the vector sum of the bird's velocity and the wind's velocity. We can use the Pythagorean theorem to find the magnitude of the resultant velocity:

$$v_{\text{resultant}} = \sqrt{v_{\text{bird}}^2 + v_{\text{wind}}^2}$$

where $v_{\text{bird}}$ is the magnitude of the bird's velocity and $v_{\text{wind}}$ is the magnitude of the wind's velocity.

Plugging in the values, we get:

$$v_{\text{resultant}} = \sqrt{(25.5 \, \text{m/s})^2 + (3.95 \, \text{m/s})^2}$$

$$v_{\text{resultant}} = \sqrt{650.25 + 15.6025}$$

$$v_{\text{resultant}} = \sqrt{665.8525}$$

$$v_{\text{resultant}} \approx 25.83 \, \text{m/s}$$

Finding the Direction of the Resultant Velocity

Now that we have the magnitude of the resultant velocity, we need to find its direction. We can use the tangent function to find the angle between the resultant velocity and the $x$-axis:

$$\tan \theta = \frac{v_{\text{wind}}}{v_{\text{bird}}}$$

where $\theta$ is the angle between the resultant velocity and the $x$-axis.

Plugging in the values, we get:

$$\tan \theta = \frac{3.95 \, \text{m/s}}{25.5 \, \text{m/s}}$$

$$\tan \theta \approx 0.155$$

Taking the inverse tangent of both sides, we get:

$$\theta \approx \tan^{-1}(0.155)$$

$$\theta \approx 8.83^\circ$$

Conclusion

In conclusion, the direction of the bird's velocity is approximately $8.83^\circ$ from the $x$-axis. This means that the bird's velocity is slightly angled in the $x$-direction due to the wind's velocity.

Discussion

This problem illustrates the concept of resultant velocity, which is the vector sum of two or more velocities. In this case, the bird's velocity and the wind's velocity are combined to produce a resultant velocity with a specific magnitude and direction. This concept is important in physics, as it helps us understand how objects move in the presence of external forces, such as wind or friction.

Applications

The concept of resultant velocity has many practical applications in fields such as aviation, navigation, and engineering. For example, pilots need to consider the wind's velocity when planning a flight route, while navigators need to take into account the wind's velocity when charting a course. Engineers also use the concept of resultant velocity when designing systems that involve moving parts, such as wind turbines or aircraft.

References

• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics (10th ed.). John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers (10th ed.). Cengage Learning.

Q: What is the resultant velocity of the bird?

A: The resultant velocity of the bird is approximately $25.83 , \text{m/s}$.

Q: What is the direction of the resultant velocity?

A: The direction of the resultant velocity is approximately $8.83^\circ$ from the $x$-axis.

Q: How does the wind's velocity affect the bird's velocity?

A: The wind's velocity affects the bird's velocity by adding a component in the $x$-direction. This means that the bird's velocity is no longer purely in the $y$-direction, but is instead angled slightly in the $x$-direction.

Q: What is the magnitude of the bird's velocity in the $y$-direction?

A: The magnitude of the bird's velocity in the $y$-direction is $25.5 , \text{m/s}$.

Q: What is the magnitude of the wind's velocity in the $x$-direction?

A: The magnitude of the wind's velocity in the $x$-direction is $3.95 , \text{m/s}$.

Q: How can we calculate the resultant velocity of the bird?

A: We can calculate the resultant velocity of the bird using the Pythagorean theorem:

$$v_{\text{resultant}} = \sqrt{v_{\text{bird}}^2 + v_{\text{wind}}^2}$$

where $v_{\text{bird}}$ is the magnitude of the bird's velocity and $v_{\text{wind}}$ is the magnitude of the wind's velocity.

Q: What is the angle between the resultant velocity and the $x$-axis?

A: The angle between the resultant velocity and the $x$-axis is approximately $8.83^\circ$.

Q: How can we use the concept of resultant velocity in real-world applications?

A: The concept of resultant velocity has many practical applications in fields such as aviation, navigation, and engineering. For example, pilots need to consider the wind's velocity when planning a flight route, while navigators need to take into account the wind's velocity when charting a course. Engineers also use the concept of resultant velocity when designing systems that involve moving parts, such as wind turbines or aircraft.

Q: What are some common mistakes to avoid when calculating resultant velocity?

A: Some common mistakes to avoid when calculating resultant velocity include:

• Failing to consider the direction of the velocities
• Failing to use the correct formula for calculating the resultant velocity
• Failing to take into account the magnitude of the velocities
• Failing to consider the effects of external forces, such as friction or air resistance

Q: How can we verify the accuracy of our calculations?

A: We can verify the accuracy of our calculations by:

• Checking our work for errors
• Using multiple methods to calculate the resultant velocity
• Comparing our results to known values or experimental data
• Considering the limitations and assumptions of our calculations

Q: What are some real-world examples of resultant velocity in action?

A: Some real-world examples of resultant velocity in action include:

• A plane flying through a headwind, which affects its velocity and direction
• A sailboat sailing through a wind, which affects its velocity and direction
• A car driving through a crosswind, which affects its velocity and direction
• A wind turbine generating electricity from the wind, which affects its velocity and direction.