A Boat Heads North Across A River At 10 Mph, While The Current Pushes It 65 ∘ 65^{\circ} 6 5 ∘ West Of North At 5 Mph. What Is The Direction Of The Boat's Resultant Vector? Measure The Angle From The Positive X X X -axis.Given:$[

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Introduction

When a boat moves across a river, it is affected by the current, which can change its direction and speed. In this scenario, a boat is heading north at a speed of 10 mph, while the current is pushing it $65^{\circ}$ west of north at a speed of 5 mph. To determine the direction of the boat's resultant vector, we need to consider the components of the boat's velocity and the current's velocity.

Understanding the Problem

The problem involves a boat moving in a two-dimensional space, with its velocity being affected by the current. We can break down the boat's velocity into its x and y components, and similarly, break down the current's velocity into its x and y components. By adding the x and y components of the boat's velocity and the current's velocity, we can determine the resultant vector of the boat's velocity.

Breaking Down the Velocities

Let's break down the boat's velocity and the current's velocity into their x and y components.

• The boat's velocity is 10 mph north, which can be broken down into:
  • x-component: 0 mph (since the boat is moving north)
  • y-component: 10 mph (since the boat is moving north)
• The current's velocity is 5 mph $65^{\circ}$ west of north, which can be broken down into:
  • x-component: $5 \cos 65^{\circ}$ mph
  • y-component: $5 \sin 65^{\circ}$ mph

Calculating the Resultant Vector

To calculate the resultant vector, we need to add the x and y components of the boat's velocity and the current's velocity.

• x-component: $0 + 5 \cos 65^{\circ} = 5 \cos 65^{\circ}$ mph
• y-component: $10 + 5 \sin 65^{\circ}$ mph

Determining the Direction of the Resultant Vector

To determine the direction of the resultant vector, we need to calculate the angle it makes with the positive x-axis. We can use the following formula:

$\theta = \tan^{-1} \left( \frac{y}{x} \right)$

where $\theta$ is the angle, $y$ is the y-component of the resultant vector, and $x$ is the x-component of the resultant vector.

Calculating the Angle

Plugging in the values, we get:

$\theta = \tan^{-1} \left( \frac{10 + 5 \sin 65^{\circ}}{5 \cos 65^{\circ}} \right)$

Using a calculator, we get:

$\theta \approx 34.5^{\circ}$

Conclusion

In conclusion, the direction of the boat's resultant vector is approximately $34.5^{\circ}$ west of north. This means that the boat will be moving in a direction that is $34.5^{\circ}$ west of the positive x-axis.

Discussion

This problem involves the concept of vectors and their components. By breaking down the boat's velocity and the current's velocity into their x and y components, we can determine the resultant vector of the boat's velocity. This is an important concept in physics, as it allows us to analyze the motion of objects in two-dimensional space.

Applications

This problem has several applications in real-life scenarios, such as:

• Navigation: Understanding the direction of a boat's resultant vector is crucial for navigation, as it helps sailors and pilots determine their course and avoid obstacles.
• Engineering: The concept of vectors and their components is used in various engineering fields, such as mechanical engineering and aerospace engineering, to analyze the motion of objects and design systems.
• Physics: This problem is a classic example of a vector problem, and it is used to teach students about the concept of vectors and their components.

References

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

Future Work

In the future, we can explore more complex vector problems, such as:

• A boat moving in a circular path, with its velocity being affected by the current.
• A particle moving in a two-dimensional space, with its velocity being affected by a force.
• A system of particles moving in a two-dimensional space, with their velocities being affected by forces.

These problems will require a deeper understanding of vectors and their components, as well as the application of mathematical techniques, such as calculus and linear algebra.

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Introduction

In our previous article, we explored the concept of vectors and their components, and how they can be used to analyze the motion of objects in two-dimensional space. We used the example of a boat moving north across a river, while the current pushes it $65^{\circ}$ west of north. In this article, we will answer some of the most frequently asked questions about this problem.

Q&A

Q: What is the direction of the boat's resultant vector?

A: The direction of the boat's resultant vector is approximately $34.5^{\circ}$ west of north.

Q: How do you calculate the resultant vector?

A: To calculate the resultant vector, you need to add the x and y components of the boat's velocity and the current's velocity. The x-component is calculated as $5 \cos 65^{\circ}$, and the y-component is calculated as $10 + 5 \sin 65^{\circ}$.

Q: What is the angle between the resultant vector and the positive x-axis?

A: The angle between the resultant vector and the positive x-axis is approximately $34.5^{\circ}$.

Q: How do you determine the direction of the resultant vector?

A: To determine the direction of the resultant vector, you need to calculate the angle it makes with the positive x-axis using the formula $\theta = \tan^{-1} \left( \frac{y}{x} \right)$.

Q: What is the significance of the resultant vector in this problem?

A: The resultant vector represents the actual velocity of the boat, taking into account the effect of the current. It is an important concept in physics, as it allows us to analyze the motion of objects in two-dimensional space.

Q: How does the resultant vector relate to the boat's motion?

A: The resultant vector represents the direction and speed of the boat's motion, taking into account the effect of the current. It is a vector quantity, which means it has both magnitude and direction.

Q: Can you provide an example of how to use the resultant vector in a real-world scenario?

A: Yes, consider a sailor who is navigating a boat across a river. The sailor needs to take into account the effect of the current on the boat's motion, and use the resultant vector to determine the actual direction and speed of the boat.

Q: What are some common applications of the resultant vector in physics?

A: The resultant vector has several applications in physics, including:

• Navigation: Understanding the direction of a boat's resultant vector is crucial for navigation, as it helps sailors and pilots determine their course and avoid obstacles.
• Engineering: The concept of vectors and their components is used in various engineering fields, such as mechanical engineering and aerospace engineering, to analyze the motion of objects and design systems.
• Physics: The resultant vector is an important concept in physics, as it allows us to analyze the motion of objects in two-dimensional space.

Conclusion

In conclusion, the resultant vector is an important concept in physics, as it allows us to analyze the motion of objects in two-dimensional space. By understanding the direction and magnitude of the resultant vector, we can determine the actual velocity of an object, taking into account the effect of external forces such as the current. This is an important concept in navigation, engineering, and physics, and has several real-world applications.

Discussion

This problem has several implications for our understanding of the motion of objects in two-dimensional space. By considering the effect of external forces such as the current, we can gain a deeper understanding of the behavior of objects in complex systems. This is an important area of research in physics, and has several real-world applications in fields such as navigation and engineering.

References

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

Future Work

In the future, we can explore more complex vector problems, such as:

• A boat moving in a circular path, with its velocity being affected by the current.
• A particle moving in a two-dimensional space, with its velocity being affected by a force.
• A system of particles moving in a two-dimensional space, with their velocities being affected by forces.

These problems will require a deeper understanding of vectors and their components, as well as the application of mathematical techniques, such as calculus and linear algebra.