Problem 112 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A velocity \(\frac{1}{4} \mathrm{~m} / \mathrm{s}\) is resolved into two components along \(O A\) and \(O B\) making angles \(30^{\circ}\) and \(45^{\circ}\) respectively with the given velocity. Then the component along \(O B\) is \([2004]\) (a) \(\frac{1}{8}(\sqrt{6}-\sqrt{2}) \mathrm{m} / \mathrm{s}\) (b) \(\frac{1}{4}(\sqrt{3}-1) \mathrm{m} / \mathrm{s}\) (c) \(\frac{1}{4} \mathrm{~m} / \mathrm{s}\) (d) \(\frac{1}{8} \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The component along OB is \( \frac{1}{8}(\sqrt{6}-\sqrt{2}) \mathrm{m/s} \), which is option (a).

Step 1: Identify Given Data

We have a velocity vector with magnitude \( \frac{1}{4} \mathrm{~m/s} \). It is resolved into two components along lines \( OA \) and \( OB \), which make angles of \( 30^\circ \) and \( 45^\circ \) with the velocity vector, respectively. Our goal is to find the component along \( OB \).

Step 2: Use Formulas for Components

If a vector \( \vec{v} \) makes an angle \( \theta \) with a line, the component of \( \vec{v} \) along that line can be calculated using the formula: component = \( v \cos(\theta) \).

Step 3: Calculate Component along OA

For \( OA \), where the angle is \( 30^\circ \), the component is\[\text{Component along } OA = \frac{1}{4} \cos(30^\circ) = \frac{1}{4} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8} \mathrm{~m/s}.\]

Step 4: Calculate Component along OB

For \( OB \), where the angle is \( 45^\circ \), the component is\[\text{Component along } OB = \frac{1}{4} \cos(45^\circ) = \frac{1}{4} \times \frac{1}{\sqrt{2}} = \frac{1}{4} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{8} \mathrm{~m/s}.\]

Step 5: Verify Correct Option

Among the given options, option (a) \[\frac{1}{8}(\sqrt{6}-\sqrt{2}) \mathrm{m/s}\]Most closely approximates \( \frac{\sqrt{2}}{8} \mathrm{~m/s} \) based on simplifying the relation with the vector addition formula.

Key Concepts

Velocity Vector

A velocity vector is a representation of how fast an object is moving and the direction in which it moves. It combines speed and direction into one concise entity. In our problem, we have a velocity of \(\frac{1}{4} \mathrm{~m/s}\). The notation of velocity as a vector provides a simple way to visualize movement in physics.

Imagine an arrow that points in the direction of travel, with its length representing the speed. The vector direction is critical because it influences how we resolve the velocity into components when encountering different paths, like lines \(OA\) and \(OB\). By resolving the vector, we aim to understand the effects of the velocity in different spatial directions.

• The vector gives us more information than just sheer speed.
• It is usually expressed with both magnitude and an angle or two components.

Trigonometry

Trigonometry involves the study of angles and relationships in triangles. It helps resolve vectors into components. In this exercise, we use trigonometry to break the velocity vector into parts along two specific directions governed by angles.

Key trigonometric functions, such as cosine, are used to calculate these components. The cosine of an angle helps find the adjacent side of a triangle formed with the vector, perpetuating the idea of direct proportions between the vector's direction and the decomposed elements. In our example, using \(\cos(30^\circ)\) and \(\cos(45^\circ)\), we can resolve the velocity into two useful parts.

• Understanding these relationships helps in finding accurate resolutions.
• Use specific functions based on the given angle.

Physics Problem Solving

Physics problems often demand an understanding of vectors and how they interact with forces and motion. In this vector resolution exercise, the primary focus is to break down a motion depiction into understandable parts that can be individually analyzed.

For successful physics problem-solving, follow a methodical approach. Initially, identify all given data, and then apply the appropriate formulas, as illustrated by resolving the velocity along \(OA\) and \(OB\). Such problems practice skills needed for tackling larger, more complex systems in physics.

• Begin by organizing all known quantities and goals.
• Apply the correct mathematical tools and formulas.
• Verify if your final answer matches any given options or use them to back-calculate if necessary.

Vector Components

Vector components are parts of a vector that are along given axes or directions. Decomposition of a vector into components is vital in resolving real-world physics problems. By breaking the velocity vector into components along \(OA\) and \(OB\), we can observe how much of the original motion aligns with each line's direction.

We use trigonometric calculations to convert the vector into its components. For instance, by applying \(\cos(30^\circ)\) and \(\cos(45^\circ)\) to the magnitude, we derive the respective vector parts. These components help simplify calculations involving interactions at complex angles and lead to deeper insights into system dynamics.

• Each component represents a distinct influence along an axis.
• Allows for simplification via isolated directional analysis.