Mass of B is four times that of A, B moves with a velocity half that of A. Then B has :

1. Basics of Motion: Velocity vs. Speed (basic)

In the study of mechanics, the most fundamental step is distinguishing between scalar and vector quantities. Speed is a scalar quantity, meaning it only describes how fast an object is moving—its magnitude. If you look at a car's speedometer, it tells you the speed at that exact moment, regardless of whether you are driving North, South, or in a circle. In contrast, Velocity is a vector quantity; it describes both how fast and in what direction an object is moving. For instance, while a jet stream might be described as having an average speed, scientists often refer to its average velocity (e.g., 120 kmph in winter) to account for its specific atmospheric flow and direction Physical Geography by PMF IAS, Jet streams, p.386.

To grasp this deeply, consider a car driving on a perfectly circular track at a constant 60 km/h. Because the number on the speedometer doesn't change, the car has a constant speed. However, because the car is constantly turning to stay on the track, its direction is always changing. Since direction is a component of velocity, the car’s velocity is changing even though its speed remains the same. This distinction is vital because in physics, a change in velocity—even if only in direction—is what defines acceleration.

Feature	Speed	Velocity
Type	Scalar (Magnitude only)	Vector (Magnitude + Direction)
Formula	Distance / Time	Displacement / Time
Can it be negative?	No, always zero or positive.	Yes, indicates opposite direction.

Understanding velocity is the gateway to more advanced concepts like work, power, and energy. For example, when we calculate the Kinetic Energy of a moving body, we use the magnitude of its velocity (speed) squared in the formula. Even though energy itself is scalar, it is the motion—the velocity—that provides the object with its capacity to do work.

Sources: Physical Geography by PMF IAS, Jet streams, p.386

2. Mass, Inertia, and Newton's Laws (basic)

In our journey to understand mechanics, we must first clear a common hurdle: the confusion between mass and weight. In everyday conversation, we use them as synonyms, but in science, they represent very different physical realities. Mass is the actual quantity of matter present in an object Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.142. It is an intrinsic property, meaning it stays the same whether you are on Earth, the Moon, or floating in deep space. Weight, however, is a force—specifically, the gravitational pull of a planet on an object Science, Class VIII NCERT, Exploring Forces, p.75. Because gravity varies from planet to planet, your weight changes depending on where you are, even though your mass remains constant.

To differentiate these concepts clearly, look at this comparison:

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Gravitational force acting on an object.
SI Unit	Kilogram (kg).	Newton (N).
Constancy	Constant everywhere in the universe.	Changes based on the local gravity.

This brings us to Inertia. Inertia is the inherent tendency of an object to resist any change in its state of rest or motion. If an object is sitting still, it wants to stay still; if it is moving, it wants to keep moving in a straight line. The "measure" of this resistance is mass. The more mass an object has, the greater its inertia. This is why it is much harder to push a stalled car than a bicycle—the car's greater mass offers a much larger resistance to changing its state of rest.

Isaac Newton synthesized these ideas into his laws of motion, marking a climax in the history of the scientific revolution Themes in world history, History Class XI NCERT, Changing Cultural Traditions, p.119. His second law gives us the famous relationship F = ma (Force = mass × acceleration). Here, the newton (N) is the standard unit used to measure force Science, Class VIII NCERT, Exploring Forces, p.65. This formula shows that if you apply the same force to two different objects, the one with more mass (more inertia) will accelerate less.

Sources: Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.142; Science, Class VIII NCERT, Exploring Forces, p.65, 75; Themes in world history, History Class XI NCERT, Changing Cultural Traditions, p.119

3. Introduction to Mechanical Energy (basic)

Mechanical energy is the energy possessed by an object due to its motion or its position. It is essentially the sum of kinetic energy (energy of motion) and potential energy (stored energy of position). In our daily lives, we see mechanical energy in action everywhere—from a wind turbine converting the kinetic energy of wind into mechanical rotation Environment, Shankar IAS Academy, Renewable Energy, p.290, to our own muscles using cellular energy (ATP) to produce the mechanical energy needed for movement Science, NCERT Class X, Life Processes, p.88.

To understand the "motion" part of mechanical energy, we look at Kinetic Energy (KE). The amount of kinetic energy an object has depends on two factors: its mass (m) and its velocity (v). The relationship is expressed by the fundamental formula:

KE = ½ mv²

This formula tells us something very important: while mass has a linear relationship with energy, velocity has a squared relationship. This means that if you double the mass, you double the energy; but if you double the velocity, the energy increases fourfold (2² = 4). Conversely, if you reduce the velocity, the energy drops significantly.

Consider a practical comparison to see how these factors balance each other out:

• Object A: Mass = m, Velocity = v. Its kinetic energy is ½ mv².
• Object B: Mass = 4m (four times heavier), but Velocity = v/2 (half as fast).

When we calculate the energy for Object B, we get: ½ × (4m) × (v/2)². Since (v/2)² is v²/4, the formula becomes ½ × 4m × v²/4. The 4s cancel out, leaving us with ½ mv². Even though Object B is much heavier, its slower speed results in the exact same kinetic energy as Object A. This demonstrates that a large increase in mass can be perfectly compensated by a relatively small decrease in velocity because of that "squared" term in the formula.

Sources: Environment, Shankar IAS Academy, Renewable Energy, p.290; Science, NCERT Class X, Life Processes, p.88

4. Linear Momentum and its Significance (intermediate)

When we talk about an object in linear motion—moving in a straight line as described in Science-Class VII, NCERT, Measurement of Time and Motion, p.116—we often focus on its speed or its mass. However, in physics, the true 'impact' of a moving object is captured by a concept called Linear Momentum. Think of momentum as the 'quantity of motion' an object possesses. It is the reason why a heavy truck moving slowly can cause as much damage as a light bullet moving very fast. Formally, linear momentum (denoted by 'p') is the product of an object's mass (m) and its velocity (v). The formula is written as p = mv. Because velocity involves both speed and direction, momentum is a vector quantity; it points in the same direction as the motion. In a straight-line path, if the speed is constant, we call it uniform linear motion Science-Class VII, NCERT, Measurement of Time and Motion, p.117, and the momentum remains constant as well. Understanding momentum is critical because it tells us how difficult it will be to stop an object. A massive object has high inertia, but once it is moving, its momentum makes it a powerhouse. To change this momentum, you must apply a force (measured in Newtons Science-Class VIII, NCERT, Exploring Forces, p.65). This leads us to a vital realization: the more momentum an object has, the more force (or time) you need to bring it to a halt.

Factor	Change	Effect on Momentum (p = mv)
Mass	Doubled (velocity constant)	Momentum doubles
Velocity	Halved (mass constant)	Momentum is reduced by half
Both	Mass doubled, Velocity halved	Momentum stays the same (2m × v/2 = mv)

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.65

5. Potential Energy and Conservation (intermediate)

In our previous discussions, we looked at energy in motion. Now, we turn to Potential Energy (PE)—the energy an object possesses not because of its movement, but because of its position or configuration. Think of it as 'stored' energy waiting to be released. The most common form we encounter in mechanics and geography is Gravitational Potential Energy (GPE). This energy is determined by three factors: the object's mass (m), the acceleration due to gravity (g), and its height (h) above a reference point. The formula is expressed simply as PE = mgh. Interestingly, because the Earth's mass is not distributed perfectly evenly, the gravitational force (and thus your potential energy) can vary slightly depending on where you are on the crust, a phenomenon known as a gravity anomaly Physical Geography by PMF IAS, Earths Interior, p.58.

The beauty of mechanics lies in the Law of Conservation of Energy. This fundamental principle states that energy cannot be created or destroyed; it can only be transformed from one form to another. In any closed system, the total energy remains constant—what goes in must balance what stays or goes out Environment and Ecology by Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. For example, a boulder perched on a cliff has high Potential Energy but zero Kinetic Energy. As it falls, its height decreases (losing PE), but its velocity increases (gaining KE). The Total Mechanical Energy (PE + KE) at any point during the fall remains the same (ignoring air resistance).

Understanding the mathematical relationship between these energies is crucial for analyzing physical systems. While Kinetic Energy (KE = ½mv²) is highly sensitive to velocity because the velocity term is squared, Potential Energy (PE = mgh) scales linearly with both mass and height. This means that if you double the height of an object, you exactly double its potential energy. However, in the case of Kinetic Energy, doubling the velocity would quadruple the energy. This interplay explains why even small changes in speed have a massive impact on the force of an impact compared to changes in mass or height alone.

Sources: Physical Geography by PMF IAS, Earths Interior, p.58; Environment and Ecology by Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14

6. The Work-Energy Theorem (intermediate)

At its heart, the Work-Energy Theorem is the bridge that connects the world of forces to the world of energy. It states that the net work done on an object by all forces acting upon it is exactly equal to the change in its Kinetic Energy (KE). In simpler terms, if you apply a force to move an object over a distance, you are transferring energy to it, which manifests as a change in its speed. As we see in environmental systems, work is done whenever one form of energy is transformed into another Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. For example, a blowing wind possesses kinetic energy that is converted into electrical energy through the work done on turbine blades INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Mineral and Energy Resources, p.61.

The formula for Kinetic Energy is KE = ½mv², where 'm' is mass and 'v' is velocity. This equation reveals a crucial insight: velocity has a much more profound impact on energy than mass does because it is squared. If you double the mass of a moving car, its energy doubles; however, if you double its speed, its energy increases fourfold. This principle explains why high-speed collisions are so much more destructive than low-speed ones, regardless of the vehicle's weight. When a force is applied against the direction of motion—such as friction when you push a box across a floor—the work done is negative, and the object's kinetic energy decreases until it comes to rest Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.67.

To master this concept for competitive exams, you must understand the balance between mass and velocity. Consider two objects: Object A has a mass 'm' and velocity 'v'. Object B is much heavier, with a mass of '4m', but moves at half the speed ('v/2'). When we calculate their energies, we find they are identical. The fourfold increase in mass for Object B is perfectly compensated by the fact that its velocity is halved (since (½)² = ¼). This mathematical symmetry is a favorite area for conceptual questions in mechanics.

Sources: Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Mineral and Energy Resources, p.61; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.67

7. Deep Dive into Kinetic Energy (KE) (exam-level)

Kinetic Energy (KE) is the energy an object possesses by virtue of being in motion. From first principles, it represents the work needed to accelerate a body of a given mass from rest to its current velocity. The mathematical expression is KE = ½mv², where 'm' is mass and 'v' is velocity. A critical nuance here is the disproportionate influence of velocity: while mass has a linear relationship with energy, velocity has a quadratic (square) relationship. This means if you double the speed of a vehicle, you don't just double its energy—you quadruple it. This concept is a cornerstone of both physics and environmental sciences. For instance, in Atmospheric Science, temperature is essentially a measure of the average kinetic energy (vibrational energy) of air molecules Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8. When water molecules gain higher kinetic energy due to temperature increases, they overcome intermolecular forces more easily, leading to increased evaporation Physical Geography by PMF IAS, Tropical Cyclones, p.358. Similarly, wind energy is the process of capturing the kinetic energy of blowing wind and converting it into electrical energy through turbines INDIA PEOPLE AND ECONOMY, NCERT 2025 ed., Mineral and Energy Resources, p.61. To understand the interplay between mass and velocity, consider the following comparison. It illustrates how a significant change in mass can be perfectly balanced by a smaller change in velocity due to the squaring effect:

Object	Mass	Velocity	KE Formula Application	Relative KE
Object A	m	v	½ × m × v²	1 KE
Object B	4m	v/2	½ × (4m) × (v/2)² = ½ × 4m × (v²/4)	1 KE

Sources: Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8; INDIA PEOPLE AND ECONOMY, NCERT 2025 ed., Mineral and Energy Resources, p.61; Physical Geography by PMF IAS, Tropical Cyclones, p.358

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental definition of Kinetic Energy (KE), this question invites you to apply the mathematical relationship where energy is determined by both mass and the square of velocity. The core concept here is understanding the proportionality; while mass affects energy linearly, velocity has a quadratic impact. As you transition from theory to PYQs, remember that UPSC often tests your ability to maintain this balance when multiple variables shift simultaneously, requiring a disciplined application of the formula NCERT Science.

Let’s walk through the logic as a coach would: if we designate the properties of object A as m and v, its energy is 1/2 mv². For object B, we substitute its specific values: a mass of 4m and a velocity of v/2. When you calculate the energy for B, the expression becomes 1/2 × (4m) × (v/2)². Crucially, squaring the half-velocity yields v²/4. When this is multiplied by the four-fold mass, the factors of four cancel out perfectly, leaving you with the original formula. This confirms that the correct answer is (A) kinetic energy equal to that of A.

To avoid common traps, look closely at the wrong options. Many students fall for Option (C) because they forget to square the velocity, simply multiplying the mass increase by the velocity decrease. Others choose Option (B) or (D) by misapplying the ratio to the entire formula. UPSC designs these distractors to catch candidates who rely on intuitive guessing rather than the squared relationship of velocity. In this specific scenario, the fourfold increase in mass is the exact compensation needed for the halving of velocity.