Understanding and Calculating Resultant Displacement Using Vectors
When analyzing the movement of an object in two-dimensional space, understanding resultant displacement is essential. Resultant displacement is the overall displacement an object experiences due to the sum of multiple individual displacements. This article will guide you through the process of calculating the resultant displacement of an object that moves 100 meters east and then 200 meters north, providing a detailed explanation and step-by-step calculation process.
Displacement Components
In this scenario, we are provided with two displacements: a 100-meter displacement (d1) towards the east and a 200-meter displacement (d2) towards the north. These displacements are represented as vector components in the x (eastward) and y (northward) directions, respectively.
East displacement: x 100 m
North displacement: y 200 m
Magnitude of the Resultant Displacement
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our scenario, the hypotenuse represents the resultant displacement (R), while the eastward and northward displacements are the two sides of the right-angled triangle.
The formula for the magnitude of the resultant displacement is:
R √(x^2 y^2)
Substituting the values:
R √(100^2 200^2)
R √(10000 40000)
R √50000 ≈ 223.61 m
Direction of the Resultant Displacement
The direction of the resultant displacement can be found using trigonometry, specifically the tangent function. The tangent of the angle (θ) is the ratio of the opposite side (y) to the adjacent side (x).
The formula to find the direction angle is:
tan θ y/x
Substituting the values:
tan θ 200/100 2
To find the angle θ:
θ arctan(2) ≈ 63.43°
The direction of the resultant displacement is therefore approximately 63.43° north of east.
Alternative Method: Law of Cosines
Alternatively, you can use the Law of Cosines to find the resultant displacement. This method is particularly useful for non-right-angled triangles. The formula for the Law of Cosines is:
R^2 d1^2 d2^2 - 2d1d2 cos(90°)
Substituting the values:
R^2 100^2 200^2 - 2 * 100 * 200 * cos(90°)
R^2 10000 40000 - 0
R^2 50000
R √50000 ≈ 223.61 m
The direction can still be found using the tangent function as described earlier.
Conclusion
Understanding and calculating resultant displacement is a fundamental concept in vector analysis. By using the Pythagorean theorem and trigonometric functions, we can accurately determine the magnitude and direction of a resultant displacement. Whether working with a right-angled triangle or not, these methods provide a reliable way to analyze and solve problems involving vector addition.