Problem 19 - A Complete Resource Book in Mathematics for JEE Main

Question

A particle starts at the origin and moves along the \(x\)-axis in such a way that its velocity at the point \((x, 0)\) is given by the formula \(\frac{d x}{d t}=\cos ^{2} \pi x\). Then the particle never reaches the point on (A) \(x=\frac{1}{4}\) (B) \(x=\frac{3}{4}\) (C) \(x=\frac{1}{2}\) (D) \(x=1\)

Step-by-Step Solution

Answer

The particle never reaches \(x = \frac{1}{2}\).

Step 1: Analyze the velocity formula

The velocity of the particle is given by \( \frac{dx}{dt} = \cos^2(\pi x) \). This expression tells us the particle's speed as it moves along the \(x\)-axis.

Step 2: Determine when velocity becomes zero

The velocity \( \frac{dx}{dt} = \cos^2(\pi x) \) equals zero when \( \cos(\pi x) = 0 \). Solving \( \cos(\pi x) = 0 \) gives \( \pi x = \frac{\pi}{2} + n\pi \) where \( n \) is any integer. Thus, \( x = \frac{1}{2} + n \).

Step 3: Identify critical points

From \( x = \frac{1}{2} + n \) where the velocity is zero, the relevant critical points on the \( x \)-axis are \( x = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots \). The particle stops at these points.

Step 4: Determine which options are critical points

The options provided are \( x = \frac{1}{4}, \frac{3}{4}, \frac{1}{2}, 1 \). The only point among these options that matches \( x = \frac{1}{2} + n \) is \( x = \frac{1}{2} \). This indicates the particle stops at \( x = \frac{1}{2} \), and cannot go beyond this point.

Key Concepts

Velocity Functions

Understanding velocity functions is essential for analyzing how particles move along a path. In our example, the velocity function is given by \( \frac{dx}{dt} = \cos^2(\pi x) \). This equation describes the velocity of a particle as it travels along the \(x\)-axis.

Velocity functions like this tell us not just the speed but also the direction of the particle at any point \((x, 0)\). In essence: - **Velocity \( \frac{dx}{dt} \) measures change**: It tells you how fast the position \(x\) changes as time goes by.- **Dimension relevance**: The velocity function is critical because it relates time to space directly when motion is only in one dimension, as in this case.
Velocity can vary based on different factors, like the presence of trigonometric functions such as cosine. These factors can make the velocity vary in a periodic manner, important to know when studying particle motion.

Critical Points in Motion

Critical points in particle motion are the values of \(x\) where something significant happens, like the particle stopping. For the velocity function \( \frac{dx}{dt} = \cos^2(\pi x) \), the critical points occur when the velocity drops to zero.
To find these points, note that \( \cos(\pi x) = 0 \) when \( \pi x = \frac{\pi}{2} + n\pi \), where \(n\) is any integer. Solving for \(x\), we get \( x = \frac{1}{2} + n \). This implies that the particle could stop at any of these points.
Typical features of critical points include:- **Location**: It's where the velocity changes, highlighting locations like \( \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots \).- **Nature**: At these points, velocity is zero, so the particle stops and cannot move past.
Identifying critical points helps predict and understand a particle's future motion. They are vital in optimization problems and analyzing systems.

Cosine Function Applications

The cosine function plays a central role in many physical and mathematical contexts, driving this exercise's velocity function. Here, it's squared to form \( \cos^2(\pi x) \). The cosine value ranges from -1 to 1, but when squared:- **Squared Nature**: \(\cos^2(\pi x)\) can only range from 0 to 1, as squaring eliminates negative values.- **Periodic Behavior**: It keeps the particle velocity periodic, affecting how the particle moves back and forth over time.
Understanding how cosine works helps solve where velocities hit zero — marking critical points. For instance, \( \cos(\pi x) = 0 \) at specific intervals, and thus apples to periodic functions where time and spatial variables repeatedly loop back. It’s important for:- **Timing**: Knowing where periodic zeroes mean determining where particles stop, crucial for predicting behavior.- **Simplicity**: Applying mathematical identities, like trigonometric identities, simplifies complex equations, offering straightforward solutions as seen in zero velocity points.
In essence, cosine functions in applications like these exemplify patterns we can predict and interpret concerning movement.