A Car Drives North For 40 Miles, Then Turns And Drives At An Angle Of $30^{\circ}$ West Of North For 40 More Miles. What Is The Magnitude Of The Car's Resultant Vector?Draw A Vector Diagram. ∣ R ⃗ ∣ = ? Miles |\vec{R}| = \, ? \text{ Miles} ∣ R ∣ = ? Miles Round Your

Mar 11, 2025 by ADMIN 269 views

A Car's Resultant Vector: A Problem in Vector Addition

Introduction

Vector addition is a fundamental concept in physics and mathematics, used to describe the motion of objects in two or more dimensions. In this article, we will explore a classic problem in vector addition, where a car drives north for 40 miles, then turns and drives at an angle of $30^{\circ}$ west of north for 40 more miles. Our goal is to find the magnitude of the car's resultant vector, which represents the total displacement of the car.

The Problem

To solve this problem, we need to break it down into smaller steps. First, we need to identify the two vectors involved: the northward vector and the westward vector. We can represent these vectors as follows:

The northward vector, denoted by $\vec{A}$, has a magnitude of 40 miles and points in the positive y-direction.
The westward vector, denoted by $\vec{B}$, has a magnitude of 40 miles and points in the negative x-direction.

Vector Addition

To find the resultant vector, we need to add the two vectors together. We can do this using the parallelogram law of vector addition, which states that the resultant vector is the diagonal of a parallelogram formed by the two vectors.

The Parallelogram Law

The parallelogram law of vector addition states that the resultant vector, denoted by $\vec{R}$, is given by:

\vec{R} = \vec{A} + \vec{B}

Using the parallelogram law, we can find the resultant vector by drawing a parallelogram with the two vectors as sides. The resultant vector is then the diagonal of the parallelogram.

Finding the Resultant Vector

To find the resultant vector, we need to use the law of cosines, which relates the magnitudes of the vectors and the angle between them. The law of cosines states that:

R^2 = A^2 + B^2 + 2AB \cos \theta

where $R$ is the magnitude of the resultant vector, $A$ and $B$ are the magnitudes of the two vectors, and $\theta$ is the angle between them.

Applying the Law of Cosines

In this case, we have:

$A = 40$ miles (the magnitude of the northward vector)$
$B = 40$ miles (the magnitude of the westward vector)$
$\theta = 30^{\circ}$ (the angle between the two vectors)$

Plugging these values into the law of cosines, we get:

R^2 = 40^2 + 40^2 + 2(40)(40) \cos 30^{\circ}

Simplifying the Equation

To simplify the equation, we need to evaluate the cosine of 30 degrees. The cosine of 30 degrees is equal to $\frac{\sqrt{3}}{2}$.

Evaluating the Cosine

The cosine of 30 degrees is equal to $\frac{\sqrt{3}}{2}$.

Substituting the Value

Substituting the value of the cosine into the equation, we get:

R^2 = 1600 + 1600 + 2(1600) \left( \frac{\sqrt{3}}{2} \right)

Simplifying the Equation

To simplify the equation, we need to multiply the terms together.

Multiplying the Terms

Multiplying the terms together, we get:

R^2 = 3200 + 1600 \sqrt{3}

Evaluating the Square Root

To evaluate the square root, we need to multiply the terms together.

Multiplying the Terms

Multiplying the terms together, we get:

R^2 = 3200 + 2743.2

Simplifying the Equation

To simplify the equation, we need to add the terms together.

Adding the Terms

Adding the terms together, we get:

R^2 = 5943.2

Finding the Magnitude

To find the magnitude of the resultant vector, we need to take the square root of both sides of the equation.

Taking the Square Root

Taking the square root of both sides of the equation, we get:

R = \sqrt{5943.2}

Evaluating the Square Root

To evaluate the square root, we need to multiply the terms together.

Multiplying the Terms

Multiplying the terms together, we get:

R = 77.0

Conclusion

Discussion

The problem of finding the resultant vector of a car's motion is a classic example of vector addition. It requires the use of the parallelogram law and the law of cosines to find the magnitude of the resultant vector. The solution to this problem is an important tool in physics and engineering, where it is used to describe the motion of objects in two or more dimensions.

Applications

The problem of finding the resultant vector of a car's motion has many applications in real-world scenarios. For example, it can be used to describe the motion of a car on a curved road, or the motion of a plane in flight. It can also be used to find the magnitude of the resultant vector of a force applied to an object, which is an important concept in physics and engineering.

Future Work

In the future, we can use this problem as a starting point to explore more complex vector addition problems. For example, we can add more vectors to the problem, or use different angles and magnitudes to find the resultant vector. We can also use this problem to explore the concept of vector addition in different coordinate systems, such as polar coordinates.

References

[1] "Vector Addition" by Khan Academy
[2] "The Parallelogram Law" by Math Open Reference
[3] "The Law of Cosines" by Math Is Fun

Code

import math
A = 40
B = 40

theta = math.radians(30)

R_squared = A2 + B2 + 2AB*math.cos(theta)

R = math.sqrt(R_squared)
print("The magnitude of the resultant vector is:", R)

Conclusion

In this article, we have used the parallelogram law of vector addition to find the resultant vector of a car's motion. We have also used the law of cosines to find the magnitude of the resultant vector. The magnitude of the resultant vector is equal to 77.0 miles. This problem has many applications in real-world scenarios, and can be used to describe the motion of objects in two or more dimensions.

Introduction

In our previous article, we explored a classic problem in vector addition, where a car drives north for 40 miles, then turns and drives at an angle of $30^{\circ}$ west of north for 40 more miles. We used the parallelogram law of vector addition to find the resultant vector, and the law of cosines to find the magnitude of the resultant vector. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the resultant vector of the car's motion?

A: The resultant vector of the car's motion is the diagonal of the parallelogram formed by the northward vector and the westward vector.

Q: How do I find the resultant vector using the parallelogram law?

A: To find the resultant vector using the parallelogram law, you need to draw a parallelogram with the two vectors as sides. The resultant vector is then the diagonal of the parallelogram.

Q: What is the law of cosines, and how is it used in this problem?

A: The law of cosines is a mathematical formula that relates the magnitudes of the vectors and the angle between them. In this problem, we used the law of cosines to find the magnitude of the resultant vector.

Q: How do I evaluate the cosine of 30 degrees?

A: The cosine of 30 degrees is equal to $\frac{\sqrt{3}}{2}$.

Q: What is the magnitude of the resultant vector?

A: The magnitude of the resultant vector is equal to 77.0 miles.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, including the description of the motion of objects in two or more dimensions. It can be used to describe the motion of a car on a curved road, or the motion of a plane in flight.

Q: Can I use this problem to explore more complex vector addition problems?

A: Yes, you can use this problem as a starting point to explore more complex vector addition problems. For example, you can add more vectors to the problem, or use different angles and magnitudes to find the resultant vector.

Q: How do I use this problem to explore the concept of vector addition in different coordinate systems?

A: You can use this problem to explore the concept of vector addition in different coordinate systems, such as polar coordinates.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Not using the parallelogram law to find the resultant vector
Not using the law of cosines to find the magnitude of the resultant vector
Not evaluating the cosine of 30 degrees correctly
Not using the correct formula for the magnitude of the resultant vector

Q: How do I check my work when solving this problem?

A: To check your work when solving this problem, you can:

Use a calculator to evaluate the cosine of 30 degrees
Use a calculator to find the magnitude of the resultant vector
Draw a diagram to visualize the vectors and the resultant vector
Check your work by plugging in the values into the formula for the magnitude of the resultant vector

Conclusion

In this article, we have answered some of the most frequently asked questions about the problem of finding the resultant vector of a car's motion. We have also provided some tips and tricks for solving this problem, and some common mistakes to avoid. By following these tips and tricks, you can ensure that you are solving this problem correctly and efficiently.

Discussion

Applications

Future Work

References

[1] "Vector Addition" by Khan Academy
[2] "The Parallelogram Law" by Math Open Reference
[3] "The Law of Cosines" by Math Is Fun

Code

import math

A = 40
B = 40

theta = math.radians(30)

R_squared = A2 + B2 + 2AB*math.cos(theta)

R = math.sqrt(R_squared)
print("The magnitude of the resultant vector is:", R)