What does the displacement vector represent in two-dimensional motion?

Question

Examoogle · Accepted Answer

The correct answer is Option B: The change in position from the initial point to the final point. The displacement vector is defined as the shortest straight-line distance between the initial position and the final position of an object. In a two-dimensional Cartesian plane, it is represented as a directed line segment pointing from the start point to the end point.

Step-by-Step Solution & Concepts:
The displacement vector is defined as the shortest straight-line distance between the initial position and the final position of an object. In a two-dimensional Cartesian plane, it is represented as a directed line segment pointing from the start point to the end point.

Working Principle: In a coordinate system, if an object moves from position $P_1(x_1, y_1)$ to position $P_2(x_2, y_2)$, the displacement vector $\vec{\Delta r}$ is calculated as the vector difference of the position vectors $\vec{r_2} - \vec{r_1}$. Since displacement is a vector quantity, it accounts for both magnitude and direction, independent of the actual path taken between the two points.

Key Formulas: $\vec{\Delta r} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$ — Representation of displacement vector in 2D cartesian coordinates; $\Delta r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ — Magnitude of the displacement vector

Key Points: Displacement is a vector quantity having both magnitude and direction.; Distance is a scalar quantity, whereas displacement is a vector.; Displacement can be zero even if the distance traveled is non-zero (e.g., circular motion).; The magnitude of displacement is always less than or equal to the actual distance traveled.

Advantages: Useful for calculating velocity vectors in kinematics.; Allows for algebraic addition of motion components using vector laws.

Disadvantages: Does not provide any information regarding the complexity or geometry of the path taken.

Applications: Navigation and GPS coordinate calculation.; Calculating average velocity in projectile motion.

Additional Info: Displacement is a state function in classical mechanics context, depending only on the endpoints.. Option A refers to the trajectory, Option C refers to the scalar length of the trajectory, and Option D is a rate calculation related to distance.

Answer: B is correct — The displacement vector signifies the direct vector change in position from the starting point to the destination.

Exam Tip: Remember that distance is always $\ge$ displacement. In competitive exams, if the motion is in a straight line without reversing direction, distance equals the magnitude of displacement.

Examoogle · Answer

Option A: The path followed by the object

Examoogle · Answer

Option C: The distance traveled along the path

Examoogle · Answer

Option D: The average speed of the object Question ID : 4410092046096 Option 1 ID : 4410098056460 Option 2 ID : 4410098056457 Option 3 ID : 4410098056459 Option 4 ID : 4410098056458 Status : Answered Chosen Option : B

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