Problem 107 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A particle has two velocities of equal magnitude inclined to each other at an angle \(\theta\). If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then \(\theta\) is \(\\{2006]\) (a) \(90^{\circ}\) (b) \(120^{\circ}\) (c) \(45^{\circ}\) (d) \(60^{\circ}\)

Step-by-Step Solution

Answer

The angle \(\theta\) is \(60^{\circ}\).

Step 1: Understanding the Problem

We have a particle with two velocities, both with the same magnitude, and an angle \( \theta \) between them. If we halve one velocity, the angle between the other original velocity and the resultant velocity is bisected by the new resultant velocity. We need to find the value of \( \theta \).

Step 2: Finding the Original Resultant Velocity

When two vectors of equal magnitude \( v \) are inclined at an angle \( \theta \), the resultant velocity \( R \) can be calculated using the formula: \[ R = \sqrt{v^2 + v^2 + 2v^2\cos\theta} = \sqrt{2v^2(1 + \cos\theta)} = v\sqrt{2(1 + \cos\theta)}. \]

Step 3: Analyzing the New System

After one velocity is halved, the new velocities are \( v/2 \) and \( v \). Let \( R' \) be the new resultant velocity with the unchanged direction. The resultant magnitude is given by: \[ R' = \sqrt{(v/2)^2 + v^2 + 2\cdot(v/2)\cdot v \cdot \cos\theta}, \] simplifying to \[ R' = \sqrt{(5/4)v^2 + v^2\cos\theta}. \]

Step 4: Bisecting the Angle

For the angle between the original resultant and the unchanged vector \( v \) to be bisected by the new resultant \( R' \), we apply the vector bisector theorem. \( \angle R \), the original angle, should be equal to \( \angle R' \). Using geometry, this scenario implies \( \tan(\theta/2) = 1/\sqrt{3} \), leading to \( \theta/2 = 30^{\circ} \). Thus, \( \theta = 60^{\circ}. \)

Step 5: Conclusion

The angle between the two velocities \( \theta \) is found to be \( 60^{\circ} \). The correct choice is (d) \( 60^{\circ}. \)

Key Concepts

Vector Addition

Understanding vector addition is essential in physics and mathematics, especially when dealing with forces, velocities, or any other vector quantity. A vector has both magnitude and direction, and the process of adding vectors involves finding a resultant vector that combines these characteristics.

• To add two vectors, you can employ the parallelogram law or the triangle method. The parallelogram law states that if two vectors are represented as adjacent sides of a parallelogram, their resultant is the diagonal of the parallelogram starting from the same point.
• The triangle method, on the other hand, involves placing the tail of the second vector at the head of the first, and the resultant vector runs from the tail of the first vector to the head of the second.

When vectors are inclined at an angle \( \theta \), the magnitude of the resultant vector can be calculated using the formula:\[ R = \sqrt{v_1^2 + v_2^2 + 2v_1v_2 \cos\theta} \]In the given problem, since both velocities have the same magnitude \( v \), the formula simplifies to \( R = v\sqrt{2(1 + \cos\theta)} \). This formula helps determine the single vector that represents the combined effect of the two given vectors.

Resultant Velocity

In physics, resultant velocity is a pivotal concept when analyzing motion composed of multiple different velocities. It is essentially a single vector that shows the combined effect of individual velocities acting simultaneously.

• To find the resultant velocity of two or more movements, you can use vector addition, as explained earlier. The direction and magnitude of the resultant velocity depend on both the magnitudes and directions of the initial velocities.
• For example, if two velocities, each with magnitude \( v \), are perpendicular, the resultant velocity follows the Pythagorean theorem resulting in a magnitude of \( \sqrt{2}v \).
• When the angle \( \theta \) between the velocities changes, so does the resultant, governed by the previously discussed formula: \( R = v\sqrt{2(1 + \cos\theta)} \).

In the problem, after halving one velocity, the new resultant velocity is recalculated accounting for the change. This adjustment allows us to analyze how the system changes with one velocity modified, further applying the angle bisector theorem to solve the exercise.

Angle Bisector Theorem

The angle bisector theorem is a geometrical tool often used to solve problems involving angles and their bisectors. This theorem helps understand how an angle is divided into two equal halves by a bisector, and it correlates the relative lengths of two segments divided by the bisector within a triangle.

• According to the angle bisector theorem, if a point divides an angle into two equal parts, then it also divides the opposite side in two segments whose lengths are proportional to the lengths of the other two sides.
• In the context of vectors and velocities, this theorem is used to relate the angles created by the resultant velocities and help find unknown angles when bisected.
• In the exercise solution, when the angle between the original resultant and one velocity is bisected by the new resultant, it means that the new resultant aligns in such a way that follows the angle bisector theorem, helping us calculate \( \theta \).

This theorem is essential for understanding how angular relationships are preserved or transformed when components like speeds or directions change, providing a deeper insight into dynamic systems.