A body is thrown vertically upwards and then falls back on the ground. Its potential energy is maximum

1. Basics of Work and Mechanical Energy (basic)

Welcome to our first step in mastering mechanics! To understand how the world moves, we must first understand Work and Mechanical Energy. In physics, Work is done when a force is applied to an object and causes it to move. This process is essentially a transfer or transformation of energy. As we see in Science, Class X (NCERT 2025 ed.), Electricity, p.188, energy must be expended to perform useful work, whether it is an electric motor turning a fan or your muscles lifting a heavy bag.

Mechanical Energy comes in two primary forms that are constantly dancing with each other:

• Kinetic Energy (KE): The energy of motion. Any object that is moving possesses KE.
• Potential Energy (PE): The stored energy an object has due to its position or configuration. The most common type we study is Gravitational Potential Energy, calculated as PE = mgh (where m is mass, g is acceleration due to gravity, and h is height).

The relationship between these two is best seen when you throw a ball vertically into the air. At the moment it leaves your hand, it has maximum speed and therefore maximum Kinetic Energy. As it rises, it slows down because gravity is doing work against it, converting that motion into height. According to Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.72, when the ball reaches its maximum height, its velocity momentarily becomes zero. At this exact point, its Kinetic Energy is zero, and its Potential Energy is at its maximum. As it falls back down, that stored PE is converted back into KE.

It is also important to note the units we use to measure these concepts. The standard SI unit for both work and energy is the Joule (J). In larger contexts, like electrical systems, we use the kilowatt-hour (kW h), where 1 kW h equals 3.6 × 10⁶ Joules Science, Class X (NCERT 2025 ed.), Electricity, p.191. Essentially, energy is never truly lost; it is simply transformed from one state to another, often dissipating as heat during the process of doing work Environment and Ecology, Majid Hussain, Basic Concepts, p.14.

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.188, 191; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.72; Environment and Ecology, Majid Hussain, Basic Concepts of Environment and Ecology, p.14

2. Forms of Energy: Kinetic and Potential (basic)

At its most fundamental level, Energy is the capacity to do work. In basic mechanics, we primarily deal with Mechanical Energy, which exists in two forms: Kinetic Energy (KE) and Potential Energy (PE). Think of Kinetic Energy as 'energy in action' and Potential Energy as 'energy in storage.' Any object in motion possesses Kinetic Energy. This is why wind can turn the heavy blades of a turbine to generate electricity—it is the energy of the moving air molecules being transferred into mechanical work Environment, Shankar IAS Academy, Renewable Energy, p.290. Even at a microscopic level, the vibration and movement of air molecules represent kinetic energy, which we often perceive and measure as temperature Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8.

Gravitational Potential Energy, on the other hand, is the energy an object possesses because of its position relative to a reference point, usually the ground. It is calculated using the formula PE = mgh, where 'm' is mass, 'g' is the acceleration due to gravity, and 'h' is the height above the reference level. This means the higher you lift an object, the more energy it stores. Gravity is a fundamental force that constantly acts upon objects, influencing their potential to fall and move Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267.

The beauty of these two forms of energy lies in their interconvertibility. Imagine throwing a ball straight up into the air. As it leaves your hand, it has maximum speed and therefore maximum Kinetic Energy. As it rises, it slows down because its KE is being converted into Potential Energy. At the very peak of its flight, the ball momentarily stops; at this precise point, its velocity is zero (meaning KE = 0) and its Potential Energy is at its maximum value Science, Class VIII NCERT, Exploring Forces, p.72. As it falls back down, that stored PE transforms back into KE, reaching maximum speed just before hitting the ground.

Sources: Environment, Shankar IAS Academy, 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; Science, Class VIII NCERT, Exploring Forces, p.72

3. The Law of Conservation of Energy (intermediate)

The Law of Conservation of Energy is a fundamental principle in physics stating that energy can neither be created nor destroyed; it can only be transformed from one form to another. In any closed system, the total energy remains constant. While we often speak of "losing" energy, in scientific terms, that energy has simply shifted into a less useful form, such as heat or sound, due to friction or air resistance Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p. 14.

In basic mechanics, we focus on the interplay between Kinetic Energy (KE)—the energy of motion—and Potential Energy (PE)—the energy of position. Imagine throwing a ball vertically upward. At the moment it leaves your hand, it has maximum KE. As it rises, it slows down because its KE is being converted into Gravitational Potential Energy. At its highest point, the ball momentarily stops: its KE becomes zero, and its PE reaches its maximum value Science, Class VIII, Chapter 5, p. 72. As it falls back down, that PE is converted back into KE. Throughout this journey, if we ignore air resistance, the sum of PE and KE stays the same.

In the real world, however, we observe that a sliding box eventually stops Science, Class VIII, Exploring Forces, p. 67. This doesn't violate the law; rather, the mechanical energy is dissipated into the environment as heat through the force of friction. This dissipation is why we must constantly provide fresh energy inputs to keep our economy and technology running, leading to an urgent global need for energy efficiency and sustainable development Contemporary India II, NCERT Class X, Print Culture and the Modern World, p. 118.

Position of Object	Kinetic Energy (KE)	Potential Energy (PE)	Total Mechanical Energy
Ground (Initial velocity)	Maximum	Zero (Reference level)	Constant
At Maximum Height	Zero (Momentary rest)	Maximum	Constant
Mid-way Point	Decreasing (Rising)	Increasing (Rising)	Constant

Sources: Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Science, Class VIII (NCERT 2025), Exploring Forces, p.72; Science, Class VIII (NCERT 2025), Exploring Forces, p.67; Contemporary India II, NCERT Class X, Print Culture and the Modern World, p.118

4. Gravitation and Acceleration due to Gravity (intermediate)

Gravitation is the universal force of attraction acting between all matter. This concept reached its scientific climax with Isaac Newton’s theory of gravitation, which posits that every mass in the universe attracts every other mass Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. On Earth, this force manifests as Weight (W), which is the product of an object's mass (m) and the acceleration due to gravity (g). While mass remains constant, weight can vary depending on the gravitational pull of the environment Science, Class VIII. NCERT (2025 ed.), Exploring Forces, p.79.

When we talk about mechanics, we must understand Gravitational Potential Energy (PE). This is the energy stored in an object due to its position or height relative to a reference point, typically the ground. It is calculated using the formula PE = mgh. Because energy is tied to height, increasing the elevation of an object increases its stored energy. For instance, a water tank placed at a greater height 'H' provides more pressure at the ground floor because the water possesses higher potential energy at that elevation Science, Class VIII. NCERT (2025 ed.), Pressure, Winds, Storms, and Cyclones, p.95.

The interplay between motion and gravity is best seen in energy transformation. When you throw a ball vertically upwards, you give it Kinetic Energy (KE). As it rises, gravity slows it down, converting that KE into PE. At the maximum height, the ball momentarily stops: its velocity is zero, meaning its Kinetic Energy is zero, and its Potential Energy is at its absolute maximum Science, Class VIII. NCERT (2025 ed.), Exploring Forces, p.72. As it falls back down, that PE is converted back into KE, illustrating the conservation of mechanical energy.

State of Motion	Kinetic Energy (KE)	Potential Energy (PE)
At Ground Level (Start)	Maximum	Zero (Reference point)
During Ascent	Decreasing	Increasing
At Maximum Height	Zero	Maximum

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII. NCERT (2025 ed.), Exploring Forces, p.72, 79; Science, Class VIII. NCERT (2025 ed.), Pressure, Winds, Storms, and Cyclones, p.95

5. Escape Velocity and Satellite Motion (exam-level)

To understand how we launch rockets or keep satellites in space, we must first look at the 'tug-of-war' between Kinetic Energy and Gravitational Potential Energy. Imagine the Earth as a deep gravity well; to climb out of it, an object needs a specific amount of energy. Escape Velocity is the minimum speed an object must reach to break free from a planet's gravitational pull forever, without further propulsion. For Earth, this is approximately 11.2 km/s. Once an object achieves this, it has enough energy to reach 'infinity' where the Earth's pull becomes negligible. This is how deep-space probes are sent to explore other planets or leave the Solar System entirely Physical Geography by PMF IAS, The Solar System, p.39.

However, if an object is moving fast but hasn't quite reached escape velocity, it may enter Satellite Motion. Instead of flying away or falling back to the ground, it falls around the Earth. This 'sideways' speed is called Orbital Velocity. The height of the orbit matters significantly: most artificial satellites are placed in the Exosphere because the air is extremely thin there, meaning there is almost no atmospheric drag to slow them down Physical Geography by PMF IAS, Earths Atmosphere, p.280. For instance, many of ISRO's monitoring satellites orbit at about 800 km altitude, circling the Earth roughly every 100 minutes Science, Class VIII (NCERT), Keeping Time with the Skies, p.185.

The relationship between these two velocities is fixed by physics: the escape velocity is always exactly √2 (about 1.41) times the orbital velocity at any given radius. This means if a satellite in a stable circular orbit suddenly increases its speed by about 41%, it will break its orbit and escape into deep space.

Feature	Orbital Velocity	Escape Velocity
Purpose	To maintain a stable path (orbit) around a planet.	To break free from a planet's gravity entirely.
Direction	Tangential (sideways) to the planet.	Any direction away from the center (usually vertical or slanted).
Energy State	Total energy is negative (the object is 'bound').	Total energy is zero or positive (the object is 'unbound').

Sources: Physical Geography by PMF IAS, The Solar System, p.39; Physical Geography by PMF IAS, Earths Atmosphere, p.280; Science, Class VIII (NCERT), Keeping Time with the Skies, p.185

6. Projectile Motion and Vertical Ascent (intermediate)

When we throw an object vertically upwards, we are witnessing a beautiful interplay between force, motion, and energy. Unlike a train moving at a constant speed, an object thrown up undergoes non-uniform linear motion because its speed is constantly changing under the influence of Earth's gravitational pull Science-Class VII, Measurement of Time and Motion, p.117. From the moment the object leaves your hand, gravity acts as a constant "braking force," causing the object to slow down until it reaches its highest point.

To understand this deeply, we must look at the Energy Transformation taking place. Every object has Potential Energy (PE), which is the energy stored due to its position, calculated as PE = mgh (where m is mass, g is gravity, and h is height). Simultaneously, its motion gives it Kinetic Energy (KE). As the object rises, its height increases, meaning its Potential Energy grows. However, because it is slowing down, its Kinetic Energy decreases. In essence, the energy is not lost; it is simply being "stored" as potential energy during the climb.

Stage of Motion	Velocity & Acceleration	Energy State
Ascent (Going Up)	Velocity decreases; direction is upward.	KE is converting into PE.
At the Peak	Velocity is zero momentarily Science, Exploring Forces, p.72.	PE is at its maximum; KE is zero.
Descent (Coming Down)	Velocity increases; direction is downward.	PE is converting back into KE.

As the object begins its journey back to Earth, the process reverses. The stored Potential Energy is released and converted back into Kinetic Energy, causing the object's speed to increase until it hits the ground Science, Exploring Forces, p.72. This cycle illustrates the Law of Conservation of Energy: the total energy (PE + KE) remains constant throughout the flight, assuming no air resistance.

Sources: Science-Class VII, Measurement of Time and Motion, p.117; Science, Exploring Forces, p.72

7. Energy Conversion in Vertical Motion (exam-level)

When we observe an object moving vertically, we are witnessing a beautiful dance between two forms of energy: Kinetic Energy (KE), the energy of motion, and Gravitational Potential Energy (PE), the energy stored due to an object's position. This storage of energy is why we place water tanks at a height; the elevated position allows the water to possess energy that can later be used to create flow and pressure Science, Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.83.

The transformation follows a predictable cycle. Imagine throwing a ball straight up into the air:

• The Ascent: As the ball leaves your hand, it has maximum velocity, meaning its Kinetic Energy is at its peak. As it rises against gravity, it slows down. This "lost" speed isn't gone; it is being converted into Potential Energy as the height (h) increases.
• The Peak: At the very top of its flight, the ball momentarily stops. Since its velocity is zero, its Kinetic Energy becomes zero. At this precise moment, the Potential Energy is at its maximum value Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.72.
• The Descent: As the ball falls back down, gravity pulls it faster and faster. The height decreases (losing PE), and that energy is converted back into Kinetic Energy until it hits the ground.

Mathematically, we define Gravitational Potential Energy as PE = mgh (where m is mass, g is acceleration due to gravity, and h is height). This relationship explains why an object's energy state is entirely dependent on its vertical position relative to the ground. In an ideal system without air resistance, the Total Mechanical Energy (KE + PE) remains constant throughout the journey.

Position	Kinetic Energy (KE)	Potential Energy (PE)
Ground Level (Launch)	Maximum	Zero (Minimum)
Highest Point	Zero	Maximum
Return to Ground	Maximum	Zero

Sources: Science, Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.83; Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.72

8. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the principles of work, energy, and motion that you have just mastered. The core building block here is the formula for Gravitational Potential Energy (PE = mgh). Since the mass (m) of the body and the acceleration due to gravity (g) are constant during the flight, the potential energy becomes a direct function of the height (h). As the body travels upward, the kinetic energy granted by the initial throw is progressively converted into potential energy, reaching its peak when the object can go no higher.

Walking through the logic, we see that as the body rises, its velocity decreases until it momentarily hits zero at the maximum height. At this specific point, the value of 'h' is at its greatest, and because the velocity is zero, the energy is no longer kinetic—it is entirely potential. According to Science, Class VIII, NCERT (Revised ed 2025), this transformation of energy is a fundamental observation of forces in motion. Therefore, the correct answer is (B), as this is the point where the displacement from the reference level is maximized.

In typical UPSC fashion, the distractors are designed to test the depth of your conceptual clarity. Option (A) is a trap because at the ground level, h = 0, making the potential energy zero. Option (C) is incorrect because during the return journey, the height is decreasing, meaning the potential energy is actually converting back into kinetic energy. Finally, option (D) is a logical contradiction; potential energy cannot be at its peak at the lowest point of a trajectory. Always remember: height is the defining variable for potential energy in these vertical motion problems.