Problem 15 - Master Resource Book in JEE Main Physics

Question

Power applied to a particle varies with time as \(P=\left(3 t^{2}-2 t+1\right)\) watt, where \(t\) is in second. Find the change in its kinetic energy between \(t=2 \mathrm{~s}\) and \(t=4 \mathrm{~s}\). (a) \(32 \mathrm{~J}\) (b) \(46 \mathrm{~J}\) (c) \(61 \mathrm{~J}\) (d) \(100 \mathrm{~J}\)

Step-by-Step Solution

Answer

The change in kinetic energy is 46 J (option b).

Step 1: Understanding the Problem

The given expression for power is a function of time: \( P = (3t^2 - 2t + 1) \). We are asked to determine the change in kinetic energy from \( t = 2 \) s to \( t = 4 \) s. As we know, power is the rate of change of kinetic energy with respect to time.

Step 2: Formula for Power and Kinetic Energy

Power \( P \) at a given time \( t \) is defined as the derivative of kinetic energy \( K \) with respect to time \( t \). Therefore, the change in kinetic energy \( \Delta K \) between two times \( t_1 \) and \( t_2 \) can be computed as the definite integral of power:\[\Delta K = \int_{t_1}^{t_2} P \, dt\]

Step 3: Setup the Integral

The integral for the change in kinetic energy is given by:\[\Delta K = \int_{2}^{4} (3t^2 - 2t + 1) \, dt\]We need to evaluate this definite integral from \( t = 2 \) to \( t = 4 \).

Step 4: Perform the Integration

Perform the integration of the polynomial function:\[\int (3t^2 - 2t + 1) \, dt = \left[ t^3 - t^2 + t \right] + C\]

Step 5: Evaluate the Definite Integral

Substitute the limits \( t = 2 \) and \( t = 4 \) into the integrated function:\[ \Delta K = \left[ (4)^3 - (4)^2 + 4 \right] - \left[ (2)^3 - (2)^2 + 2 \right]\]Calculating these, we find:For \( t = 4 \):\( 64 - 16 + 4 = 52 \)For \( t = 2 \):\( 8 - 4 + 2 = 6 \)Thus, \( \Delta K = 52 - 6 = 46 \) J.

Key Concepts

Power as a Function of Time

In physics, power is an important concept that describes how fast energy is being used or transferred at any moment. When power is expressed as a function of time, it shows how the power changes as time progresses. In our exercise, the power applied to a particle is given by the expression \( P = (3t^2 - 2t + 1) \) where \( t \) represents time in seconds.

This means that at any second \( t \), you can determine the power by substituting the value of \( t \) into the expression. For instance, at \( t = 2 \) seconds, the power is \( P = (3(2)^2 - 2(2) + 1) \) watts. Understanding how power varies over time is crucial in determining how energy is transferred to or from the system.

• Power tells us the rate of energy change.
• Functions like \( P(t) \) allow us to model real-world scenarios.
• Helps relate the concept of work done over time in physical systems.

Definite Integral

The definite integral is a fundamental concept in calculus used to calculate the total accumulation of quantities, such as area under a curve or the change in energy over a time interval. In the context of our problem, the definite integral determines the total change in kinetic energy over a specified time period.

When given a power function \( P(t) \), the change in kinetic energy \( \Delta K \) from \( t = 2 \) to \( t = 4 \) seconds can be expressed using the definite integral: \[ \Delta K = \int_{2}^{4} (3t^2 - 2t + 1) \, dt \] This integral calculates the accumulated power, which equates to the change in kinetic energy.

• Definite integrals provide exact values over intervals.
• They are key to connecting how quantities change over time.
• Useful for translating continuous functions into real-world quantities, such as energy.

Kinematics

Kinematics is the branch of mechanics that deals with motion without considering the forces that cause it. It often focuses on quantities such as displacement, velocity, and acceleration. In terms of energy, kinetic energy is connected to the motion of objects through velocity.

In the problem given, we are exploring changes in kinetic energy, which is directly linked to the kinematic properties of a particle. Although we do not calculate velocity directly here, understanding that kinetic energy is \( KE = \frac{1}{2}mv^2 \) is crucial because the energy changes as the square of velocity changes.

• Kinematics helps describe how an object moves.
• Provides foundations to understand kinetic energy relationships.
• Covers aspects like speed, critical for energy calculations.