Interpreting Particle Movement and Displacement Along the Line x y

In the field of physics, understanding particle movement and displacement is foundational for solving complex problems in mechanics. This article explores a specific case: a particle moving from the origin with a velocity of v 3i 4j and investigates its displacement after two seconds along the line x y. This problem highlights the importance of vector operations and the application of geometric principles in solving physics questions.

Understanding the Problem

Given that a particle moves with a velocity of v 3i 4j starting from the origin, we are tasked with finding its displacement after two seconds along the line x y. Let's break down this process step-by-step.

Step 1: Determining the Particle's Position After Two Seconds

First, we need to calculate the particle's position after two seconds. This is achieved through the formula for displacement:

Displacement Velocity × Time

Given:

Substituting these values into the formula:

Displacement (3i 4j) × 2 6i 8j

Therefore, the position of the particle after two seconds is 6i 8j.

Step 2: Determining the Displacement Along the Line x y

To find the displacement of the particle along the line x y, we need to project the position vector (6i 8j) onto the line x y. This involves finding the component of the displacement vector along the line x y.

Step 2.1: Representing the Line x y

The line x y can be represented by the vector d i - j.

Step 2.2: Finding the Closest Point on the Line x y

The closest point on the line x y to the point (6, 8) can be determined by finding the values of t in the equation x y t. The point is (7, 7).

Step 2.3: Calculating the Displacement Along the Line x y

The displacement from the point (6, 8) to (7, 7) is:

Displacement (7i - 7j) - (6i 8j) i - 1j

Therefore, the displacement of the particle along the line x y is d i - j.

Alternative Method: Using the Dot Product

Another approach to solve the problem involves using the dot product to find the component of the displacement vector along the line x y

The unit vector along the line x y is u (1/sqrt(2)) (i - j).

The component of displacement along the line x y can be found as:

Component of displacement (1/sqrt(2)) (i - j) ? (6i 8j) (1/sqrt(2)) (6 - 8) (-2/sqrt(2)) -sqrt(2)

The resultant vector along the line x y is thus -sqrt(2) (i - j), or -sqrt(2)i sqrt(2)j.

This method provides an alternative way to verify the initial solution, showing the importance of vector operations in analytical physics.

Conclusion

Understanding particle movement and displacement, especially when constrained by geometric conditions like x y, is crucial in various applications of physics and engineering. The use of vector algebra and geometric projections allows us to solve complex problems systematically and accurately. This problem exemplifies the interplay between different mathematical concepts and their application in solving practical physics questions.