Problem 29 - Master Resource Book in JEE Main Physics

Question

A \(0.5 \mathrm{~kg}\) ball is thrown up with an initial speed \(14 \mathrm{~ms}^{-1}\) and reaches a maximum height of \(8 \mathrm{~m} .\) How much energy is dissipate by air drag acting on the ball during the ascent? (a) \(19.6 \mathrm{~J}\) (b) \(4.9 \mathrm{~J}\) (c) \(10 \mathrm{~J}\) (d) \(9.8 \mathrm{~J}\)

Step-by-Step Solution

Answer

The energy dissipated by air drag is 9.8 J.

Step 1: Determine Initial Kinetic Energy

The initial kinetic energy (KE) of the ball is given by the formula:\[ KE_{initial} = \frac{1}{2}mv^2 \]Substitute the given values:\[ KE_{initial} = \frac{1}{2} \times 0.5 \times (14)^2 \]\[ KE_{initial} = \frac{1}{2} \times 0.5 \times 196 \]\[ KE_{initial} = 49 \text{ J} \]

Step 2: Calculate Potential Energy at Maximum Height

The potential energy (PE) at the maximum height can be calculated using the formula:\[ PE = mgh \]where \( g = 9.8 \text{ m/s}^2 \).Substitute the given values:\[ PE = 0.5 \times 9.8 \times 8 \]\[ PE = 39.2 \text{ J} \]

Step 3: Calculate Energy Dissipated by Air Drag

The energy dissipated by air drag (E_d) is the difference between the initial kinetic energy and the potential energy at the maximum height:\[ E_d = KE_{initial} - PE \]Substitute the values calculated:\[ E_d = 49 \text{ J} - 39.2 \text{ J} \]\[ E_d = 9.8 \text{ J} \]

Step 4: Identify the Correct Answer

From the calculated energy dissipated by air drag, the correct answer is:(d) \(9.8 \text{ J}\)

Key Concepts

Understanding Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It depends on both the mass of the object and its velocity. The formula for kinetic energy is given by:\[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object.
• \( v \) is the velocity of the object.

In this scenario, a ball is thrown upwards with a certain speed. Its initial kinetic energy can be calculated using the formula above. As the ball moves up, its speed reduces due to the force of gravity and other forces, converting its kinetic energy into potential energy.

This transformation continues until the ball reaches its highest point where its velocity is zero, meaning all its kinetic energy has been converted. At this point, the ball has the maximum potential energy and zero kinetic energy.

Exploring Potential Energy

Potential energy is the energy stored in an object due to its position or state. For gravitational potential energy, which is applicable here, the focus is on the height of an object relative to a reference point. The potential energy can be calculated using the formula:\[ PE = mgh \]where:

• \( m \) is the mass of the object.
• \( g \) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2 \) on Earth).
• \( h \) is the height of the object above the reference point.

As the ball ascends, it gains height, hence increasing its potential energy. By the time the ball reaches its maximum height, potential energy is at its highest and equals the initial kinetic energy, subtracting any energy lost to air resistance. This illustrates the conservation of energy, where energy is transferred from kinetic to potential form.

The Role of Air Drag

Air drag is a force that opposes the motion of an object through air, leading to energy dissipation. When an object moves, it must push against the air molecules in its path, which takes away some energy in the form of heat and sound.

In the case of the ball being thrown upwards, not all the kinetic energy is converted to potential energy. Some energy is lost due to air drag, which causes the ball to not reach an unlimited height. The energy dissipated by air drag can be calculated by finding the difference between the initial kinetic energy and the potential energy at the height reached:\[ E_d = KE_{initial} - PE \]This energy quantity showcases the amount of energy lost to air drag during the motion of the ball. Understanding this loss is critical as it demonstrates why objects do not move indefinitely under real-world conditions. In our exercise, this dissipation accounts for a significant part of the energy transition and highlights the practical impact of air drag.