91Ó°ÊÓ

In physics, the velocity function describes how the speed and direction of an object change over time. In this problem, the velocity function is given as \( v(t) = 4 - 4 \sin \left( \frac{\pi}{2} t \right) \). This function specifies the velocity of a particle moving along a line, providing insights into its motion. The velocity function's form \( 4 - 4 \sin \left( \frac{\pi}{2} t \right) \) suggests an oscillatory component due to the sine function.

• \( 4 \) represents a constant velocity component.
• \(-4 \sin \left( \frac{\pi}{2} t \right) \) oscillates between \(-4\) and \(4\), representing a varying part of the velocity.

Understanding this function is crucial, especially when it comes to studying periodic motion or wave phenomena, which commonly appear in physics, particularly in oscillatory systems.

The definite integral is a fundamental concept in calculus that calculates the area under a curve over a specified interval. It’s vital for finding quantities that accumulate over time, such as distance traveled by an object. To find the average velocity, we need to compute the definite integral of the velocity function over the given interval.
The formula for average velocity \( V_{avg} \) using definite integrals is: \[ V_{avg} = \frac{1}{b-a} \int_{a}^{b} v(t) \, dt \]
Here, \( a \) is the initial time, and \( b \) is the end time. With our problem's bounds \( a = 0 \) and \( b = (2n-1)2 \), we evaluate the integral:

• First, for the constant term: \( 4 \), resulting in \( 8(2n-1) \).
• Second, for the oscillatory sine term, which evaluates to zero over a complete cycle.

Combining these results provides the average velocity. Integrals like these are potent tools for handling complex physics problems and are also indispensable in solving JEE physics challenges.

Trigonometric substitution is an effective technique used to evaluate integrals involving trigonometric functions. It simplifies calculations by transforming complex expressions into more manageable forms. In our problem, to evaluate the integral of \( 4 \sin \left( \frac{\pi}{2} t \right) \), we introduce a substitution: \( u = \frac{\pi}{2} t \). This offers a new differential \( dt = \frac{2}{\pi} du \), simplifying the integrals' limits and variables.
Here are the steps involved in this substitution:

• Change of variable: \( u = \frac{\pi}{2} t \), transforming the integral's variable.
• New differential: \( dt = \frac{2}{\pi} du \) allows for integration regarding \( du \).
• Updated limits follow from the initial bound transformations: \( t = 0 \rightarrow u = 0 \) and \( t = (2n-1)2 \rightarrow u = (2n-1) \pi \).

By applying this method, the sine integral resolves smoothly, allowing the overall average velocity calculation to be executed more effectively.

Physics problems for JEE (Joint Entrance Examination) encompass a broad spectrum of conceptual and application-based challenges. Mastery of various physical concepts like velocity, kinematics, and integration is essential. This problem, related to the average velocity, is emblematic of the type of inquiries students face.

• JEE problems demand a thorough understanding of underlying principles, such as motion equations and calculus applications.
• Adaptability in techniques, like trigonometric substitution, ensures students' capability to address complex integrals common in physics.
• Such questions enhance problem-solving skills, enabling students to tackle real-world scientific queries.

Tackling JEE-level physics problems aids in building a strong foundational knowledge. This knowledge is paramount for advancing further into science and engineering curricula.