A rubber ball dropped from 24 m height loses its kinetic energy by 25%. What is the height to which it rebounds?

1. Introduction to Work and Mechanical Energy (basic)

In our journey to master mechanics, we must first understand that Energy is the fundamental 'currency' of the physical world. It is broadly defined as the capacity to do work. As noted in NCERT, Contemporary India II, Print Culture and the Modern World, p.113, energy is required for every activity we see around us—from cooking and lighting to propelling vehicles and driving industrial machinery. In physics, Work is specifically defined as the process of energy transfer that occurs when a force moves an object over a distance. If you apply force but there is no movement, scientifically speaking, no work has been done.

When we zoom into Mechanical Energy, we are looking at the sum of two specific types of energy: Kinetic Energy (KE) and Potential Energy (PE). Kinetic energy is the energy of motion. For instance, wind turbines harness the kinetic energy of moving air to rotate blades, which is then converted into mechanical power for tasks like grinding grain or generating electricity Environment Shankar IAS, Renewable Energy, p.290. On the other hand, Potential Energy is 'stored' energy based on an object's position or state—like a stretched rubber band or an object held at a height.

A crucial principle to remember is that energy is never truly 'lost' in a closed system; it only changes form. However, in the real world, not all energy is converted into useful work. For example, when an electric fan rotates, some electrical energy is converted into the mechanical work of turning blades, while some is dissipated as heat or sound Science class X, Electricity, p.188. Understanding this balance between potential energy (position), kinetic energy (motion), and work (transfer) is the bedrock of all mechanical concepts.

Sources: NCERT, Contemporary India II, Print Culture and the Modern World, p.113; Environment Shankar IAS, Renewable Energy, p.290; Science class X, Electricity, p.188

2. Gravitational Potential Energy (PE) (basic)

Gravitational Potential Energy (GPE) is the energy an object possesses because of its position in a gravitational field. In simple terms, it is "stored energy" that has the potential to be converted into other forms, like motion. When you lift an object against the pull of gravity, you are doing work on it, and that work is stored as GPE. This gravitational force is a fundamental constant in our lives, influencing everything from the rotation of the Earth to the movement of tectonic plates Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267.

The amount of GPE an object has depends on three primary factors, expressed in the formula: PE = mgh

• m (Mass): The heavier the object, the more energy it stores.
• g (Acceleration due to gravity): On Earth, this is roughly 9.8 m/s². It is important to note that 'g' isn't perfectly uniform everywhere; differences in the density of materials within the Earth's crust create gravity anomalies, which slightly alter the gravitational pull in different locations Physical Geography by PMF IAS, Earths Interior, p.58.
• h (Height): The higher an object is lifted from a reference point (usually the ground), the greater its potential energy.

When an object is released and starts falling, its GPE begins transforming into Kinetic Energy (KE)—the energy of motion. In a perfect scenario, all the GPE would convert to KE. However, in real-world mechanics, when an object like a ball hits the ground, some of that energy is "lost" to the environment. It doesn't disappear; it simply changes into non-mechanical forms like heat, sound, or the energy required to slightly deform the object. This is why a bouncing ball rarely returns to its original height—it has lost a fraction of its total energy during the impact.

Sources: Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267; Physical Geography by PMF IAS, Earths Interior, p.58

3. Kinetic Energy (KE) and Conservation (intermediate)

At its simplest level, Kinetic Energy (KE) is the energy an object possesses due to its motion. Whether it is a planet revolving around the sun or a molecule vibrating in the atmosphere, if it is moving, it has kinetic energy. The mathematical expression for this is KE = ½mv², where 'm' represents mass and 'v' represents velocity. This formula tells us two vital things: KE increases linearly with mass, but it increases quadratically with speed. This is why a vehicle doubling its speed actually quadruples its kinetic energy, making it much harder to stop Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.119.

The Law of Conservation of Energy states that energy cannot be created or destroyed; it only changes form. In a perfect vacuum, a falling object converts its Potential Energy (PE = mgh) entirely into Kinetic Energy. However, in the real world, we deal with 'energy transformations' where some energy leaves the primary system. For instance, wind turbines convert the kinetic energy of wind into mechanical energy and then into electricity Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.290. When a moving object strikes a surface, not all KE is preserved as motion; some is converted into sensible heat (molecular vibration) or sound Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8.

Understanding this 'energy budget' is crucial for the UPSC, as it applies to everything from Geothermal energy to the Rotation of the earth Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267. When we say an object 'loses' energy, we usually mean it has been dissipated as heat or sound due to friction or inelastic impact. If a ball loses 25% of its energy upon hitting the ground, it simply means that 75% remains available to be converted back into potential energy for the rebound.

Type of Energy	Definition	Transformation Example
Potential (PE)	Energy of position or state.	A ball held at a height 'h'.
Kinetic (KE)	Energy of motion.	The ball falling at velocity 'v'.
Thermal/Heat	Random KE of molecules.	Energy lost during impact or friction.

Sources: Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.119; Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.290; Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267

4. Connected Concept: Friction and Air Resistance (intermediate)

In our journey through mechanics, we must meet the 'reluctant partner' of motion: Friction. Friction is a force that resists the relative motion between two surfaces in contact. While we often think of it as a nuisance that wears out our shoes, it is fundamental to how the world functions. When surfaces rub against each other, the microscopic irregularities interlock, requiring energy to overcome. This process leads to the dissipation of energy—meaning kinetic energy is not lost from the universe, but rather converted into less useful forms like heat and sound. Even the Earth itself experiences this; the gravitational pull of the moon causes tidal flexing, which creates internal friction, slowly converting the Earth’s rotational energy into heat and causing our days to lengthen by a tiny fraction over centuries Physical Geography by PMF IAS, Earth's Interior, p.59.

When an object moves through a gas or a liquid, it encounters a specific type of friction known as Fluid Friction or Air Resistance (Drag). Unlike friction between solid surfaces, which is relatively constant regardless of speed, air resistance increases significantly as an object moves faster. This is because the object must physically push aside more air molecules per second. For example, as an air parcel rises or falls due to density changes, it interacts with its environment, involving complex energy exchanges Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.297. In UPSC preparation, it is vital to remember that in any real-world mechanical system, some energy will always be 'lost' to the environment as thermal energy due to these resistive forces.

Feature	Solid Friction	Air Resistance (Drag)
Medium	Between two solid surfaces.	Between a solid and a fluid (air/gas).
Dependence	Depends on the nature of surfaces.	Depends on speed and surface area.
Energy Outcome	Converted mostly to heat/wear.	Converted mostly to heat/turbulence.

Sources: Physical Geography by PMF IAS, Earths Interior, p.59; Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.297

5. Connected Concept: Newton’s Laws of Motion (intermediate)

To understand why objects move the way they do, we must look at the three pillars of classical mechanics: Newton’s Laws of Motion. At their core, these laws describe how forces interact with mass to create motion. We start with the concept of Inertia (First Law), which states that an object will maintain its state of rest or uniform linear motion unless an external force intervenes. Think of a train moving along a straight track; it requires a force to start, and a force (brakes/friction) to come to a halt Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. Without these forces, the object would simply keep doing what it is already doing.

The Second Law provides the mathematical backbone: F = ma (Force = mass × acceleration). This tells us that the Newton (N), our SI unit for force, is essentially the amount of push or pull required to accelerate a specific mass Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.65. In the case of a ball thrown upward, the force of gravity acts against its motion, causing it to slow down, stop momentarily at the peak, and then accelerate downward as gravity continues to pull it toward the Earth's center Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.78.

Finally, the Third Law explains interactions: for every action, there is an equal and opposite reaction. When a ball hits the ground, it exerts a downward force on the floor; the floor simultaneously exerts an upward force on the ball, causing it to rebound. While the forces are equal, the energy is not always conserved in the same form. During such an impact, some energy is lost to heat or sound, which is why a ball typically doesn't bounce back to its original height. This framework of motion and gravity was the crowning achievement of the scientific revolution, moving us from mere observation to precise prediction Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119.

Law	Key Concept	Real-world Application
First Law	Inertia	A passenger jerking forward when a bus stops suddenly.
Second Law	Acceleration (F=ma)	It is easier to push an empty shopping cart than a full one.
Third Law	Action/Reaction	The recoil of a gun after firing a bullet.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116; Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.65; Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.78; Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119

6. Elastic vs. Inelastic Collisions (exam-level)

In mechanics, a collision occurs when two or more bodies exert forces on each other in a relatively short time. While the Law of Conservation of Momentum applies to all collisions in a closed system, they are categorized into two types based on whether Kinetic Energy (KE) is conserved: Elastic and Inelastic collisions.

In an Elastic Collision, the total kinetic energy of the system remains the same before and after the impact. No energy is transformed into heat, sound, or deformation. A classic example is the collision between subatomic particles or, approximately, the strike of billiard balls. Conversely, in an Inelastic Collision, kinetic energy is not conserved. While the objects might still move, some of their initial KE is converted into other forms of energy such as heat, sound, or internal potential energy due to the permanent deformation of the objects.

To understand this in a practical context, consider a rubber ball dropped from a height (h). As it falls, its potential energy (PE = mgh) converts into kinetic energy. If the collision with the floor were perfectly elastic, the ball would bounce back to the exact same height. however, in the real world, a ball typically slows down or stops because energy is lost to the environment Science, Class VIII, Exploring Forces, p.78. If a ball loses, say, 25% of its kinetic energy during the impact, it retains only 75% (0.75) of its energy to rebound. Consequently, its new peak height (h') will be 75% of the original height (h' = 0.75h).

Feature	Elastic Collision	Inelastic Collision
Momentum	Conserved	Conserved
Kinetic Energy	Conserved	Not Conserved (Converted to heat/sound)
Real-world Example	Subatomic particles	A ball bouncing on the floor, car crashes

Sources: Science, Class VIII, NCERT, Exploring Forces, p.78

7. Mathematical Relationship: Energy and Rebound Height (exam-level)

To understand why a ball doesn't return to its original height after a bounce, we must look at the Law of Conservation of Energy and the reality of inelastic collisions. When an object is held at a height (h), it possesses Gravitational Potential Energy (PE), calculated as PE = mgh (where 'm' is mass and 'g' is gravity). As the ball falls, this potential energy transforms into Kinetic Energy (KE). In a theoretical "perfect" vacuum with a perfectly elastic floor, the ball would retain 100% of its energy and return to height 'h'.

However, in the real world, energy is "lost" during the impact. This energy isn't destroyed but is converted into other forms such as thermal energy (heat), acoustic energy (sound), and the work required for the elastic deformation of the ball's material. If a ball loses a certain percentage of its energy (e.g., 25%), it means it retains the remainder (75%) to move back upward. Because height is a linear component of the potential energy formula, the relationship is direct: if you have 75% of your energy left, you will reach 75% of your previous height. Calculating such dimensions accurately is a basic requirement in physical sciences Science, Class VIII NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.145.

Mathematically, we equate the energy after impact to the potential energy at the new peak height (h'):

• Energy Retained = (Retention Factor) × mgh
• mgh' = (Retention Factor) × mgh
• By cancelling 'mg' from both sides, we find: h' = (Retention Factor) × h

This linear relationship allows us to predict rebound heights across multiple bounces, provided the percentage of energy loss remains constant per impact.

Sources: Science, Class VIII NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.145

8. Solving the Original PYQ (exam-level)

This question beautifully synthesizes the principles of Work, Energy, and Power that you have just mastered. By applying the Law of Conservation of Energy, you can visualize the ball's journey: its initial Potential Energy (PE) at 24m is converted into Kinetic Energy (KE) upon impact. The "25% loss" described is a practical application of inelastic collisions, where energy is dissipated through heat and sound. As you learned in NCERT Class 9 Science, since PE is calculated as mgh, the height is directly proportional to the total mechanical energy available to the ball.

To arrive at the correct answer, (C) 18 m, walk through the logic step-by-step: if the ball loses 25% of its energy, it retains 75% (or 0.75) of its original energy for the rebound. Since mass and gravity remain constant, the rebound height ($h'$) is simply 75% of the initial height ($h$). Calculating 0.75 × 24 (or $3/4$ of 24) leads you directly to 18 m. Thinking in terms of ratios and proportions rather than complex kinematic equations is a vital skill for the UPSC CSAT, as it saves precious time during the exam.

UPSC often includes "distractor" options to catch students who read too quickly. Option (A) 6 m is a classic trap; it represents the 25% lost height ($0.25 × 24$), rather than the height reached by the remaining energy. Option (D) 24 m assumes a perfectly elastic collision with zero energy loss, which contradicts the problem statement. Finally, (B) 12 m is a common guess for students who instinctively pick a halfway point without performing the calculation. Always focus on what the question asks for—the resultant state, not the change itself.