Calculating Total Displacement A Car's Journey East And North

In the fascinating realm of physics, displacement is a fundamental concept that describes the change in an object's position. Unlike distance, which measures the total path traveled, displacement focuses solely on the net change in position from the starting point to the ending point. This concept is crucial in understanding motion and is particularly relevant in fields like navigation, mechanics, and engineering. In this article, we will delve into a practical problem involving a car's displacement, exploring how to calculate the total displacement when the car travels in different directions. We will break down the problem step-by-step, applying vector addition principles to arrive at the solution. This exercise will not only enhance our understanding of displacement but also illustrate how vector concepts are applied in real-world scenarios. So, let's embark on this journey of understanding displacement and unravel the complexities of motion.

Problem Statement

Imagine a scenario where a car embarks on a journey, first traveling 30 meters eastward, and then changing direction to travel 40 meters at an angle of 45° north of east. Our objective is to determine the car's total displacement. This means we need to find the magnitude and direction of the car's overall change in position from its starting point. To solve this, we'll employ the principles of vector addition, as displacement is a vector quantity, possessing both magnitude and direction. The challenge lies in combining these two displacements, which occur in different directions, to find the resultant displacement. This problem exemplifies how vector addition is used in real-world situations to calculate the net effect of multiple movements or forces acting on an object. By breaking down each displacement into its horizontal and vertical components, we can systematically add them together to find the resultant displacement vector. So, let's dive into the solution and discover the car's total displacement.

Breaking Down the Problem: Vector Components

To effectively tackle this problem, we need to decompose the second displacement (40 meters at 45° north of east) into its horizontal (eastward) and vertical (northward) components. This is a crucial step in vector addition, as it allows us to treat each direction independently and then combine them to find the resultant vector. Using basic trigonometry, we can find these components. The eastward component of the 40-meter displacement is given by 40 * cos(45°), and the northward component is given by 40 * sin(45°). Since cos(45°) and sin(45°) are both equal to √2/2 (approximately 0.707), we can calculate these components. The eastward component is approximately 40 * 0.707 = 28.28 meters, and the northward component is also approximately 40 * 0.707 = 28.28 meters. Now, we have the second displacement broken down into its eastward and northward components, which makes it easier to combine with the first displacement. This decomposition method is a fundamental technique in physics for handling vectors, allowing us to simplify complex problems into manageable parts. In the next step, we will add these components to find the total displacement.

Vector Addition: Calculating Total Displacement

Now that we have the components of the second displacement, we can proceed with vector addition to find the total displacement. The car's first displacement was 30 meters east, and the second displacement has components of 28.28 meters east and 28.28 meters north. To find the total eastward displacement, we add the eastward components of both displacements: 30 meters + 28.28 meters = 58.28 meters. The total northward displacement is simply the northward component of the second displacement, which is 28.28 meters, as there was no northward displacement in the first part of the journey. Now, we have the total eastward and northward displacements, which are the components of the total displacement vector. To find the magnitude of the total displacement, we use the Pythagorean theorem: √(58.28² + 28.28²). This calculation gives us the magnitude of the total displacement, which represents the straight-line distance from the starting point to the ending point. Additionally, we need to find the direction of the total displacement, which can be determined using trigonometry. The angle (θ) of the displacement north of east can be found using the arctangent function: θ = arctan(28.28 / 58.28). This angle gives us the direction of the car's total displacement. By calculating both the magnitude and direction, we can fully describe the car's total displacement from its initial position.

Magnitude and Direction of Total Displacement

Let's calculate the magnitude and direction of the total displacement using the values we've obtained. The magnitude of the total displacement is calculated using the Pythagorean theorem: √((58.28 m)² + (28.28 m)²) = √(3396.56 m² + 799.75 m²) = √(4196.31 m²) ≈ 64.78 meters. This is the straight-line distance from the car's starting point to its final position. To find the direction, we calculate the angle θ using the arctangent function: θ = arctan(28.28 / 58.28) ≈ arctan(0.485) ≈ 25.88 degrees. This angle represents the direction of the total displacement north of east. Therefore, the car's total displacement is approximately 64.78 meters at an angle of 25.88 degrees north of east. This result gives us a complete understanding of the car's change in position, considering both the distance and direction traveled. The magnitude tells us how far the car is from its starting point, and the angle tells us the direction in which the car has moved relative to the east direction. This comprehensive analysis showcases the power of vector addition in solving real-world problems involving displacement.

Conclusion: Significance of Displacement in Physics

In conclusion, we have successfully determined the car's total displacement by breaking down the problem into vector components, adding them together, and calculating the magnitude and direction of the resultant vector. The car's total displacement was found to be approximately 64.78 meters at an angle of 25.88 degrees north of east. This exercise highlights the importance of understanding vector addition and its applications in physics, particularly in analyzing motion. Displacement, as a vector quantity, provides a more complete picture of an object's change in position compared to distance, which is a scalar quantity. Displacement considers both the magnitude and direction of the change, making it a crucial concept in fields such as mechanics, navigation, and engineering. Understanding displacement allows us to predict and analyze the motion of objects in various scenarios, from simple movements to complex trajectories. The principles we've applied here can be extended to solve a wide range of problems involving vector quantities, reinforcing the fundamental role of vector analysis in physics. By mastering these concepts, we gain a deeper understanding of the physical world and the principles that govern motion.