Problem 71 - Master Resource Book in JEE Main Physics

Question

A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time \(t\) is proportional to [NCERT] (a) \(t^{12}\) (b) \(t\) (c) \(t^{3 / 2}\) (d) \(t^{2}\)

Step-by-Step Solution

Answer

The power delivered is proportional to \( t \), so the answer is (b).

Step 1: Understand the Problem

We are given a scenario where a body is initially at rest and undergoes one-dimensional motion with constant acceleration. The task is to determine how the power delivered to it varies with time, given several options.

Step 2: Define the Physical Quantities

Let the initial velocity be zero, i.e., \( u = 0 \). The acceleration is constant, denoted by \( a \). Power \( P \) is defined as \( P = F \cdot v \), where \( F \) is force and \( v \) is velocity. We will express each quantity in terms of time \( t \).

Step 3: Use Equation of Motion

Given the initial velocity is zero, from the equation of motion, the velocity after time \( t \) can be written as: \[ v = u + at = at \] Since \( u = 0 \), it simplifies to \( v = at \).

Step 4: Express Force in Terms of Mass and Acceleration

The force \( F \) acting on the body changing its velocity is given by Newton's second law: \[ F = ma \]where \( m \) is the mass of the body.

Step 5: Write Power in Terms of Known Quantities

Power delivered to the body is given by: \[ P = F \cdot v \] Substitute \( F = ma \) and \( v = at \) into the power equation: \[ P = ma \cdot at = ma^2t \] Thus, power \( P \) is directly proportional to \( t \).

Step 6: Conclusion

After analyzing the relationship, we find that the power delivered to the body is proportional to time \( t \). Thus, the correct option is (b) \( t \).

Key Concepts

Constant Acceleration

Constant acceleration occurs when an object's velocity changes at a consistent rate over time. This means the object's speed increases by the same amount in equal intervals of time. In many exercises in mechanics, constant acceleration is assumed to simplify the problem.
Under constant acceleration, there are a few key equations that are often used to analyze motion. One of the fundamental equations is \[ v = u + at \]where:

• \( v \) is the final velocity.
• \( u \) is the initial velocity.
• \( a \) is the constant acceleration.
• \( t \) is the time elapsed.

This equation allows us to calculate how fast an object is moving at any point in time, given its starting speed and the constant acceleration. It's a building block for solving many physics problems, especially those involving power which relates force and motion.

Newton's Second Law

Newton's Second Law of Motion is a cornerstone of classical mechanics. It establishes the relationship between an object's mass, its acceleration, and the force applied to it. The law is succinctly expressed as:\[ F = ma \]where:

• \( F \) is the force acting on an object.
• \( m \) is the mass of the object.
• \( a \) is the acceleration of the object.

This formula tells us that the force applied to an object is equal to the mass of the object multiplied by the acceleration produced.
In practical terms, it means that for a given force, heavier objects (larger mass) will accelerate less than lighter objects under the same force.
Understanding this relationship is crucial when examining power, as power involves the rate at which work is done or energy is transferred by these forces over time.

Equation of Motion

Equations of motion are mathematical expressions that describe the behavior of a moving body in terms of its velocity, position, acceleration, and time. They are particularly useful in understanding one-dimensional motion.
The fundamental equations of motion in one dimension when starting with constant acceleration include:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

Where:

• \( v \) is the final velocity.
• \( s \) is the displacement.
• \( t \) is the time.
• \( u \) is the initial velocity.
• \( a \) is the acceleration.

These equations allow us to calculate various parameters of motion. They are used to determine an object's current state or predict where it will be in the future, given certain initial conditions.
These tools are invaluable when analyzing mechanical problems, like determining how power changes with time in the context of motion.

One-Dimensional Motion

One-dimensional motion refers to movement that occurs along a straight line, either horizontally or vertically. In such motions, an object's position changes along a single axis, simplifying the analysis.
In mechanics, the study of one-dimensional motion often involves examining how an object's velocity and position change over time due to forces. Key considerations include:

• Initial velocity: the speed and direction of an object’s motion at the starting point.
• Final velocity: speed and direction of an object after a certain period of time.
• Displacement: the change in position from the starting point to the endpoint.

In problems involving constant acceleration, like a car accelerating along a road or an object falling under gravity, one-dimensional motion provides a fundamental framework to predict and analyze the movement.
By focusing on a single direction, complex multi-directional movements can be broken down into more manageable components, simplifying problem-solving in fields such as physics or engineering.