If a light body and a heavy body have equal momentum, then

1. Understanding Mass, Velocity, and Inertia (basic)

To understand mechanics, we must first distinguish between Mass and Weight—two terms often used interchangeably in daily life but treated very differently in science. Mass is defined as the actual quantity of matter present in an object Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.142. It is an intrinsic property, meaning it does not change regardless of where the object is located. In contrast, Weight is the gravitational force with which a planet pulls that object toward itself Science, Class VIII, Exploring Forces, p.75. While your mass remains the same on Earth or the Moon, your weight would change because the gravitational pull is different.

Next, we consider Velocity and Inertia. Velocity is not just how fast something moves (speed), but the speed of an object in a specific direction. Even in space, velocity varies; for instance, the Earth's orbital speed increases as it nears the sun and decreases as it moves away Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257. Inertia is the inherent tendency of an object to resist any change in its state of rest or motion. This resistance is directly tied to mass: the more mass an object has, the greater its inertia. It is much harder to push a stationary truck than a stationary bicycle because the truck’s larger mass provides significantly more inertia.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Force of gravity acting on an object.
S.I. Unit	Kilogram (kg)	Newton (N) Science, Class VIII, Exploring Forces, p.65
Variability	Constant everywhere.	Changes with gravity/location.

Sources: Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.142; Science, Class VIII, Exploring Forces, p.75; Science, Class VIII, Exploring Forces, p.65; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257

2. Linear Momentum: The Quantity of Motion (basic)

When we talk about an object moving in a straight line, we are discussing linear motion. As you might recall from your early science studies, an object moving along a straight track—like a train between two stations—is the simplest form of motion to analyze (Science-Class VII, Measurement of Time and Motion, p.116). However, to truly understand how that object interacts with the world, we need a concept called Linear Momentum (represented by the symbol p). Often described as the "quantity of motion," momentum is the product of an object's mass (m) and its velocity (v). The formula is written as p = mv.

While momentum and Kinetic Energy (K) both depend on mass and speed, they represent different physical realities. Momentum is a measure of how difficult it is to stop a moving object, while Kinetic Energy is the energy the object possesses due to its motion. These two are mathematically linked by the formula K = p²/2m. This relationship reveals something fascinating: if two objects have the same momentum, their kinetic energies will not be the same. Instead, the kinetic energy becomes inversely proportional to the mass (K ∝ 1/m).

To visualize this, imagine a heavy truck and a light bicycle both having the exact same momentum. Because the bicycle has much less mass, it must be traveling at a much higher velocity to maintain that same level of momentum (since p = mv). Because Kinetic Energy depends on the square of the velocity (K = ½mv²), that extra speed gives the lighter object a much higher "energy score." In any scenario where momentum is held constant, the lighter body will always possess more kinetic energy than the heavier one.

Sources: Science-Class VII, Measurement of Time and Motion, p.116; Science-Class VIII, Exploring Forces, p.77

3. Kinetic Energy and the Work-Energy Theorem (basic)

Welcome back! Now that we understand motion, let’s talk about Kinetic Energy (K)—the energy an object possesses due to its motion. From a first-principles perspective, any mass (m) moving with a velocity (v) carries this energy, defined by the classic formula K = ½mv². We see this principle in action everywhere, from wind turbines converting the kinetic energy of air into electricity Environment, Shankar IAS Academy, Renewable Energy, p.290, to the propulsion of vehicles across distances Contemporary India II, NCERT, Energy Resources, p.113. Even at a microscopic level, the temperature we feel is actually a measure of the vibrational kinetic energy of molecules Environment and Ecology, Majid Hussain, Basic Concepts, p.8.

To truly master this for competitive exams, you must understand the deep link between Kinetic Energy and Momentum (p). Momentum is the "quantity of motion" an object has, calculated as p = mv. If we rearrange these variables, we find a very elegant relationship: K = p² / 2m. This formula tells us that if two objects have the same momentum, their kinetic energies will depend entirely on their masses. Specifically, because mass (m) is in the denominator, kinetic energy is inversely proportional to mass (K ∝ 1/m) when momentum is held constant.

Imagine a heavy truck and a light bicycle moving with the exact same momentum. For the bicycle to match the truck's momentum, it must be traveling at a much higher velocity. Because the kinetic energy formula squares the velocity (v²), that high speed gives the lighter bicycle significantly more kinetic energy than the heavy truck. This is a counterintuitive but vital concept in mechanics!

Scenario (Constant Momentum)	Mass (m)	Velocity (v)	Kinetic Energy (K)
Heavy Body	High	Low	Lower
Light Body	Low	High	Higher

Sources: Environment, Shankar IAS Academy, Renewable Energy, p.290; Contemporary India II, NCERT, Energy Resources, p.113; Environment and Ecology, Majid Hussain, Basic Concepts, p.8

4. Potential Energy and Conservation Laws (intermediate)

To master mechanics, we must look beyond objects in motion and consider the energy stored within them due to their position or configuration—what we call Potential Energy (PE). Think of a rock perched on a cliff; it isn't moving, but it has the 'potential' to do work because of its height in Earth's gravitational field. This force of gravity is a fundamental driver of terrestrial processes, varying based on mass and position—for instance, gravity measurements in deep oceanic trenches show variations that indicate a loss or displacement of mass Physical Geography by PMF IAS, Tectonics, p.108. On a cosmic scale, these gravitational interactions are so powerful that they can distort the very fabric of spacetime Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5.

The bridge between potential and movement is the Law of Conservation of Energy. This law dictates that energy cannot be created or destroyed, only transformed. When work is done, one form of energy is converted into another—like potential energy turning into kinetic energy as an object falls Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. In a closed system, the total energy remains constant. This principle is not just a physics rule; it governs the circulation of matter and energy throughout our biosphere, ensuring that while energy may dissipate as heat, the total 'input' and 'output' within the system remain balanced.

A critical nuance for UPSC students is the relationship between Momentum (p) and Kinetic Energy (K). While momentum is mass multiplied by velocity (p = mv), kinetic energy is half the mass times the square of velocity (K = ½mv²). By combining these, we get the formula: K = p²/2m. This mathematical relationship reveals a counter-intuitive truth: if two objects—one light and one heavy—have the exact same momentum, the lighter object will have more kinetic energy. This is because the lighter object must travel at a much higher velocity to reach the same momentum as the heavy one, and since velocity is squared in the energy equation, it contributes more significantly to the total energy.

Sources: Physical Geography by PMF IAS, Tectonics, p.108; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5; Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14

5. Power and Mechanical Efficiency (intermediate)

To understand Power, think of it as the 'speed' of energy consumption or work. While work tells us how much energy was transferred, power tells us how fast it happened. In the administrative world, a policy that achieves a goal in six months is more 'powerful' than one that takes six years, even if the final result (the work done) is identical. Mathematically, Power (P) = Work (W) / Time (t). The SI unit of power is the Watt (W), which is equivalent to one joule per second (1 J/s) Science, class X (NCERT 2025 ed.), Electricity, p.192.

In practical applications, we often deal with larger scales. For instance, the energy we pay for in our homes is measured in Kilowatt-hours (kWh). This is a unit of energy, not power! One kWh represents the total energy consumed when a 1000-watt appliance runs for one hour, which equals 3.6 × 10⁶ Joules Science, class X (NCERT 2025 ed.), Electricity, p.191. Whether we are discussing the insolation (solar power) received by the Earth—which varies from 320 W/m² in the tropics to 70 W/m² at the poles—or industrial machinery, power defines the capacity for rapid action Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Solar Radiation, Heat Balance and Temperature, p.68.

Mechanical Efficiency, on the other hand, is a measure of how well a system converts input energy into useful work. No machine is 100% efficient because energy is always lost to heat or friction. Efficiency (η) is the ratio: (Useful Output Work / Total Input Energy) × 100%. A fascinating nuance arises when comparing bodies of different masses. If a light body and a heavy body have the same momentum (p), the lighter body actually possesses more Kinetic Energy (K). This is because K = p²/2m; with 'p' being constant, a smaller mass 'm' leads to a higher energy 'K'. To match the momentum of a heavy object, the light object must travel at a much higher velocity. Since energy increases with the square of velocity (½mv²), the high-speed lighter object ends up carrying significantly more energy to manage!

Sources: Science, class X (NCERT 2025 ed.), Electricity, p.191-192; Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Solar Radiation, Heat Balance and Temperature, p.68

6. Collisions and Conservation of Momentum (exam-level)

In our study of linear motion — which occurs when an object moves along a straight path (Science-Class VII, Measurement of Time and Motion, p.116) — we often distinguish between uniform motion (constant speed) and non-uniform motion (changing speed) (Science-Class VII, Measurement of Time and Motion, p.117). However, to truly master mechanics, we must look at how mass and velocity combine to form momentum (p), defined as p = mv. Momentum is a measure of the "quantity of motion" and is fundamentally conserved in closed systems during collisions. While momentum is proportional to velocity (p = mv), Kinetic Energy (K) is proportional to the square of velocity (K = ½mv²). We can link these two vital concepts using the formula K = p²/2m. This relationship reveals a fascinating truth: if a heavy body and a light body have the same momentum, the lighter body will always possess more kinetic energy. Physically, this happens because for a lighter body to achieve the same momentum as a heavy one, it must travel at a much higher velocity. Since kinetic energy is sensitive to the square of that velocity, the speed advantage of the light body outweighs its mass disadvantage, resulting in higher energy.

Feature	Heavy Body (Mass M)	Light Body (Mass m)
Momentum (p)	Equal (p)	Equal (p)
Velocity (v)	Lower	Higher
Kinetic Energy (K)	Lower (K = p²/2M)	Higher (K = p²/2m)

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

7. The Mathematical Link: Momentum vs. Kinetic Energy (exam-level)

To master mechanics, we must understand how Momentum (p) and Kinetic Energy (K) relate to one another. While both concepts describe a body in motion, they prioritize mass and velocity differently. Momentum is a linear measure of motion (p = mv), while Kinetic Energy represents the scalar energy an object possesses due to that motion (K = ½mv²). As we know from Science, Class VIII, Exploring Forces, p.64, applying a force can change an object's speed, which simultaneously alters both these values. However, the most critical insight for competitive exams is the mathematical bridge between them: K = p²/2m.

This formula is derived by substituting the definition of velocity (v = p/m) into the kinetic energy equation. It reveals a vital relationship: if two objects have the same momentum, their kinetic energies are inversely proportional to their masses (K ∝ 1/m). This leads to a counter-intuitive but essential physical truth: when a light body and a heavy body move with equal momentum, the lighter body possesses more kinetic energy. This happens because the lighter body must travel at a much higher velocity to compensate for its low mass and match the momentum of the heavier object. Since kinetic energy increases with the square of velocity, that high speed outweighs the lower mass in the energy calculation.

Consider the comparison below for two objects with identical momentum (p):

Feature	Light Body (Small m)	Heavy Body (Large m)
Velocity (v)	Very High	Low
Kinetic Energy (K)	Higher (since K = p²/2m)	Lower (since K = p²/2m)

Sources: Science, Class VIII (NCERT Revised ed 2025), Exploring Forces, p.64

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental definitions of Momentum (p) and Kinetic Energy (K), this question brings those building blocks together. The key is to utilize the derived relationship K = p²/2m, which bridges the gap between these two concepts. In the UPSC syllabus, moving from individual definitions to understanding the interdependence of variables—like how mass (m) influences energy when momentum is constant—is exactly where the examiner tests your conceptual depth.

To arrive at the correct answer, we observe that because momentum (p) is the same for both bodies, the Kinetic Energy becomes inversely proportional to mass (K ∝ 1/m). Reasoning logically: for a light body to match the momentum of a heavy body, it must possess a much higher velocity. Since Kinetic Energy depends on the square of the velocity (v²), this high speed results in a significantly larger energy output. Thus, (A) the lighter body has greater kinetic energy than the heavier body is the only logically sound conclusion.

UPSC often includes options like (B) and (C) to exploit common misconceptions. Option (B) is a "heavy means more" trap for students who ignore the velocity factor, while (C) targets those who mistakenly believe that equal momentum implies an equal energy state. Option (D) is a classic distractor meant to see if you can be swayed into thinking the relationship is arbitrary. As a civil services aspirant, always look for the mathematical proportionality rather than relying on surface-level intuition.