Problem 58 - Master Resource Book in JEE Main Physics

Question

A body of mass \(2 \mathrm{~kg}\) is thrown up vertically with kinetic energy of \(490 \mathrm{~J}\). The height at which the kinetic energy of the body becomes half of its original value is (a) \(50 \mathrm{~m}\) (b) \(12.25 \mathrm{~m}\) (c) \(25 \mathrm{~m}\) (d) \(10 \mathrm{~m}\)

Step-by-Step Solution

Answer

The correct height is approximately 12.25 m, thus the answer is (b) 12.25 m.

Step 1: Understand the Problem

We need to find the height at which the kinetic energy of a 2 kg mass, initially with 490 J of kinetic energy, becomes half its original value. The problem involves concepts of kinetic and potential energy.

Step 2: Write the Expression for Kinetic Energy

The kinetic energy (KE) of the body initially is 490 J. We aim to find the height at which KE becomes half, i.e., \( \frac{490}{2} = 245 \) J.

Step 3: Express Energy Conservation

According to the conservation of energy principle, the total energy remains constant. At height \( h \), the potential energy (PE) gained is \( 490 - 245 = 245 \) J, because the loss in kinetic energy becomes potential energy.

Step 4: Use the Potential Energy Formula

The potential energy at height \( h \) is given by \( PE = mgh \), where \( m \) is the mass (2 kg), \( g \) is the gravitational acceleration (9.8 m/s²), and \( h \) is the height. Set \( 2gh = 245 \) J.

Step 5: Solve for Height

Substitute \( m = 2 \) kg and \( g = 9.8 \text{ m/s}^2 \) in the equation: \( 2 \times 9.8 \times h = 245 \). So, \( 19.6h = 245 \), implying \( h = \frac{245}{19.6} \approx 12.5 \) m.

Key Concepts

Kinetic Energy Calculations

Kinetic energy is the energy that an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy (KE) is given by \[ KE = \frac{1}{2}mv^2 \] where \( m \) represents the mass of the object and \( v \) is its velocity.
In this particular problem, we started with a kinetic energy of 490 J for a 2 kg mass. Kinetic energy being a scalar quantity indicates that only the speed of the object matters, not the direction.

• When kinetic energy is discussed, think about how fast an object is moving.
• Be watchful of unit conversion; consistency is key.
• Doubling the speed of an object quadruples its kinetic energy due to the square relationship with velocity.

By understanding kinetic energy, one can predict how an object will behave while in motion, such as how high it will travel if thrown upward.

Potential Energy Formula

Potential energy is stored energy due to an object's position or configuration. For objects at a certain height, we often refer to gravitational potential energy (PE). This form of energy depends on the object's mass, the height from the reference point, and the gravitational constant. The basic formula for gravitational potential energy is \[ PE = mgh \]where \( m \) stands for mass, \( g \) for gravitational acceleration (typically \(9.8 \text{ m/s}^2\)), and \( h \) is the height.
When an object is lifted to a height, it gains potential energy due to the work done against gravity.

• Think of potential energy as the energy held because of an object's location.
• More height means more potential energy if all other factors are constant.
• In problems like these, potential energy conversion is key to solving for unknowns such as height.

The height where the kinetic energy of our object was halved resulted in a specific potential gain, illustrating the exchange between potential and kinetic energy.

Energy Conversion in Mechanics

In mechanics, energy conversion is a cornerstone concept that helps explain the behavior of systems. The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. This principle was crucial in solving the given problem.
Initially, the body had kinetic energy of 490 J. As it ascended, some kinetic energy converted into potential energy. At the height where its kinetic energy was halved, the remaining portion converted into potential energy.

• Conservation of energy helps track changes in energy forms to solve for unknowns.
• In this problem, the full conversion is captured through \( KE + PE = \text{constant} \).
• Knowing the amount of energy conversion aids in understanding object behavior in motion.

Energy conversion doesn't just apply to this simple mechanics question. It's a fundamental concept in various fields, from engineering to everyday applications, making it invaluable for analyzing and predicting system outcomes.