Problem 18 - Master Resource Book in JEE Main Physics

Question

A particle of mass \(2 \mathrm{~kg}\) starts moving in a straight line with an initial velocity of \(2 \mathrm{~ms}^{-1}\) at a constant acceleration of \(2 \mathrm{~ms}^{-2}\). Then rate of change of kinetic energy (a) is four times the velocity at any moment (b) is two times the displacement at any moment (c) is four times the rate of change of velocity at any moment (d) is constant through out

Step-by-Step Solution

Answer

The rate of change of kinetic energy is four times the velocity (option a).

Step 1: Understand the Given Problem

The problem involves a particle with a given mass, initial velocity, and constant acceleration. We need to determine how the rate of change of its kinetic energy relates to other physical quantities like velocity, displacement, and acceleration.

Step 2: Recall the Formula for Kinetic Energy

The kinetic energy \( KE \) of a particle with mass \( m \) and velocity \( v \) is given by \( KE = \frac{1}{2}mv^2 \).

Step 3: Differentiate Kinetic Energy with Respect to Time

To find the rate of change of kinetic energy, differentiate the kinetic energy formula with respect to time:\[ \frac{d}{dt}\left(\frac{1}{2}mv^2\right) = m v \frac{dv}{dt} \]Since \( \frac{dv}{dt} = a \) (acceleration), this becomes:\[ \frac{d(KE)}{dt} = m v a \]

Step 4: Plug in the Given Values

Given \( m = 2 \) kg and \( a = 2 \) m/s², substitute these into the equation for rate of change of kinetic energy:\[ \frac{d(KE)}{dt} = 2 \cdot v \cdot 2 = 4v \]

Step 5: Analyze the Rate of Change of Kinetic Energy

From the derivation, we can see that the rate of change of kinetic energy is \( 4v \), which suggests that it is four times the velocity at any moment. Thus, option (a) is the correct answer.

Key Concepts

Differentiation in Physics

Differentiation is a fundamental tool in physics, especially when dealing with motion. In the context of kinetic energy, differentiation helps us understand how this energy changes over time. If you recall, kinetic energy is calculated as \( KE = \frac{1}{2}mv^2 \). This equation tells us how much energy a moving object possesses due to its motion. However, to know how quickly this energy changes, we must differentiate it.By differentiating kinetic energy with respect to time, we obtain its rate of change. Using the chain rule of differentiation, we express:

• \( \frac{d(KE)}{dt} = m v \frac{dv}{dt} \)

This equation reveals that the change involves both the change in velocity \( \frac{dv}{dt} \), which is acceleration, and the current velocity \( v \). Differentiation, thus, becomes an essential process in analyzing physical systems where variables depend on time.

Newton's Laws of Motion

Newton's Laws of Motion provide critical insights into how objects move and interact with forces. These laws are the foundation of classical mechanics, describing the relationship between the motion of an object and the forces acting upon it. In the context of our problem:

• **First Law**: An object at rest stays at rest, and an object in motion stays in motion at a constant velocity unless acted upon by an external force.
• **Second Law**: This is articulated as \( F = ma \), indicating that force is the product of mass and acceleration. This principle helps us understand how acceleration, force, and velocity are interconnected.
• **Third Law**: For every action, there is an equal and opposite reaction.

In our exercise, the particle's acceleration is given, aligning with the second law. The constant acceleration implies a constant force is acting on it. This law helps us to derive the rate of change of kinetic energy since acceleration explains how velocity alters with time, hence influencing kinetic energy.

Kinematics Equations

Kinematics is the study of motion without considering the forces that cause it. The kinematic equations describe how objects move under constant acceleration. For a particle moving in a straight line, the main equations are:

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

where \( u \) is the initial velocity, \( v \) is the final velocity, \( a \) is acceleration, \( t \) is time, and \( s \) is displacement.In this problem, these equations help us verify how the velocity of the particle changes with time due to constant acceleration. Knowing the velocity at any point lets us calculate kinetic energy and, through differentiation, its rate of change. These equations guide us in understanding the velocity and displacement of an object as functions of time, and relate back to the kinetic energy equation we've been discussing.