Two bodies A and B having masses m and 4m respectively are moving with equal linear momentum. The ratio of kinetic energies between A and B is

1. Foundations of Motion: Mass and Velocity (basic)

To understand how things move, we must first master two fundamental building blocks: Mass and Velocity. While we often use these terms casually, science requires us to be precise. Mass is the measure of the 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 does not change regardless of where the object is located. Whether you are on Earth, the Moon, or floating in deep space, your mass remains the same because the amount of "stuff" you are made of hasn't changed.

It is crucial to distinguish Mass from Weight. While mass is measured in kilograms (kg), weight is actually a force measured in Newtons (N) Science Class VIII NCERT, Exploring Forces, p.75. Weight is the result of gravity pulling on that mass. Because gravity varies across the universe, your weight can change, but your mass is constant.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Force of gravitational attraction.
SI Unit	Kilogram (kg)	Newton (N)
Variability	Constant everywhere.	Changes with gravity/location.

Now, let's look at Velocity. In simple terms, velocity is speed with a specific direction. When an object moves along a straight line at a constant speed, we call it uniform linear motion Science Class VII NCERT, Measurement of Time and Motion, p.117. If the speed changes or the direction shifts, the motion becomes non-uniform. Interestingly, velocity isn't just for solid objects; in geography, we see that temperature differences between air masses determine the velocity of jet streams—the greater the temperature contrast, the faster the velocity Physical Geography by PMF IAS, Jet streams, p.385.

Sources: Science Class VIII NCERT, Exploring Forces, p.75; Science Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.142; Science Class VII NCERT, Measurement of Time and Motion, p.117; Physical Geography by PMF IAS, Jet streams, p.385

2. Newton’s Second Law and the Concept of Force (basic)

To understand how objects move, we must first look at Mass—the quantity of matter present in an object. Unlike weight, which can change depending on gravity, mass remains constant no matter where you are in the universe Science - Class VIII, Exploring Forces, p.75. When this mass starts moving along a straight line, we call it Linear Motion Science - Class VII, Measurement of Time and Motion, p.116. Newton’s Second Law provides the bridge between force and this motion by introducing Momentum (p), which is simply the product of an object's mass (m) and its velocity (v), expressed as p = mv.

While momentum describes the "oomph" or quantity of motion an object carries, Kinetic Energy (K) describes the energy it possesses due to that motion. These two concepts are deeply intertwined. If you know an object's momentum and its mass, you can calculate its kinetic energy using the formula: K = p² / 2m. This formula is a powerful tool in mechanics because it allows us to compare the energy of different objects without always needing to know their specific speeds.

Consider a fascinating scenario: what happens if a heavy truck and a light bicycle have the exact same momentum? For their momenta to be equal, the lighter bicycle must be traveling at a much higher velocity than the heavy truck. Because velocity is "squared" in energy calculations, that extra speed gives the lighter object significantly more kinetic energy. In simple terms, if momentum is held constant, the lighter the body, the higher its kinetic energy.

Sources: Science - Class VIII, Exploring Forces, p.75; Science - Class VII, Measurement of Time and Motion, p.116

3. Linear Momentum: Definition and Units (intermediate)

In our previous discussions, we explored how objects move along a straight line, which we call linear motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. However, to truly understand the 'strength' of that motion, we need the concept of Linear Momentum. Think of momentum as the 'quantity of motion' contained in a body. It isn't just about how fast an object is moving, but also how much 'stuff' (mass) is moving. This is why a heavy truck moving slowly can be much harder to stop than a light bicycle moving at the same speed. Formally, Linear Momentum (p) is defined as the product of an object's mass (m) and its velocity (v). The mathematical expression is p = mv. It is crucial to remember that momentum is a vector quantity, meaning it has both magnitude and a specific direction — the same direction as the velocity of the object. Whether the motion is uniform (constant speed) or non-uniform (changing speed) Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117, the momentum at any given instant is always determined by the instantaneous mass and velocity. To determine the units of momentum, we look at its components. We know from our study of measurements that the SI unit of mass is the kilogram (kg) and the SI unit of speed or velocity is the metre per second (m/s) Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. By multiplying these together, we find that the SI unit of linear momentum is kg m/s (kilogram-metre per second). In smaller laboratory scales, you might also see the CGS unit, which is g cm/s (gram-centimetre per second), similar to how density can be expressed in different scales depending on the context Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113, 116, 117; Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141

4. Connected Concept: Conservation of Momentum (intermediate)

When we talk about Momentum (represented by the symbol p), we are describing the "quantity of motion" an object possesses. It is the product of an object's mass (m) and its velocity (v), expressed as p = mv. While linear motion describes an object moving in a straight line—like a train moving between two stations Science-Class VII, Measurement of Time and Motion, p.116—momentum tells us how difficult it would be to stop that object. A force is required to change this state of motion, whether to speed it up, slow it down, or change its direction Science, Class VIII, Exploring Forces, p.64.

The Law of Conservation of Momentum states that in an isolated system (where no external forces act), the total momentum remains constant. This principle is why rockets, like those launched from the Thumba station, can propel themselves forward by ejecting fuel backward at high speeds Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.78. However, a deeper layer of this concept is how momentum relates to Kinetic Energy (K). While both involve mass and velocity, they weigh these factors differently. We can link them using the formula K = p² / (2m).

This relationship leads to a non-intuitive but vital conclusion: If two objects have the exact same momentum, the lighter object will always have more kinetic energy. This is because, to maintain the same momentum as a heavy object, the lighter one must move significantly faster. Since kinetic energy is proportional to the square of the velocity (v²), that extra speed boosts the energy much more than the lower mass reduces it. This comparison is summarized below:

Scenario (Equal Momentum)	Mass (m)	Kinetic Energy (K = p²/2m)
Object A (Heavy)	High	Low
Object B (Light)	Low	High

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.64; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.78

5. Connected Concept: Work-Energy Theorem (intermediate)

To understand the Work-Energy Theorem, we must first look at how we define "work" and "energy" in a physical sense. In simple terms, work is the measure of energy transfer that occurs when an object is moved over a distance by an external force. In our earlier steps, we saw how potential difference moves charges Science, class X (NCERT 2025 ed.), Electricity, p.173; similarly, in mechanics, a net force acting on a body does work, which results in a change in the body's Kinetic Energy (K)—the energy of motion.

The Work-Energy Theorem states that the net work done by all forces acting on a particle equals the change in its kinetic energy. Mathematically, if we apply a force to an object of mass m, its velocity changes from u to v, and the work done is expressed as: W = ΔK = ½mv² - ½mu². This principle is fundamental because it allows us to calculate the effects of forces without needing to track the exact acceleration at every micro-second. We see the macroscopic results of this energy in nature, such as when wind or running water possesses the kinetic energy necessary to erode and transport earth materials FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.43.

A crucial intermediate step for competitive exams is understanding the relationship between Kinetic Energy (K) and Linear Momentum (p). Momentum is defined as the product of mass and velocity (p = mv). By manipulating the kinetic energy formula, we can establish a direct link:

• Start with: K = ½mv²
• Multiply and divide by m: K = (m²v²) / (2m)
• Since p = mv, then p² = m²v²
• Therefore: K = p² / (2m)

This formula reveals a vital conceptual insight: at a constant momentum, kinetic energy is inversely proportional to mass (K ∝ 1/m). This means if two objects have the same momentum, the lighter object will actually possess significantly more kinetic energy because it must travel at a much higher velocity to maintain 그 match in momentum.

Scenario (Constant Momentum)	Mass (m)	Kinetic Energy (K)
Lighter Body	Low	High
Heavier Body	High	Low

Sources: Science, class X (NCERT 2025 ed.), Electricity, p.173; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.43

6. Relating Kinetic Energy to Momentum (exam-level)

To understand the relationship between Kinetic Energy (K) and Linear Momentum (p), we must first look at their standard definitions from first principles. Momentum is the measure of the quantity of motion an object possesses, defined as the product of its mass (m) and velocity (v), or p = mv. On the other hand, Kinetic Energy is the energy an object possesses due to its motion, expressed as K = ½mv². While we often treat these as separate metrics, they are mathematically tethered through the object's mass. By rearranging the momentum formula to solve for velocity (v = p/m) and substituting it into the kinetic energy equation, we derive a powerful new relationship: K = p² / (2m). This formula is incredibly useful in competitive exams because it allows us to compare the energy of two bodies without knowing their individual velocities. For instance, in atmospheric studies, we see that the kinetic energy of molecules (expressed as temperature) is a direct result of their motion, though the density and mass of those molecules dictate how that energy is perceived as sensible heat Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8. The most critical takeaway from this relationship is how mass influences the outcome when momentum is held constant. Since mass (m) is in the denominator (K ∝ 1/m), if two objects have the same momentum, the lighter object will always possess more kinetic energy. Conversely, if two objects have the same kinetic energy, the heavier object will possess greater momentum. This inverse relationship is a cornerstone of classical mechanics often tested in UPSC conceptual physics questions.

Condition	Resulting Relationship	Interpretation
Constant Momentum (p)	K ∝ 1/m	The lighter the body, the higher its Kinetic Energy.
Constant Kinetic Energy (K)	p ∝ √m	The heavier the body, the higher its Momentum.

Sources: Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8

7. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the fundamental relationship between Momentum and Kinetic Energy. Having mastered the individual definitions of mass, velocity, and energy, you are now applying the bridge formula K = p² / 2m. In the UPSC context, these problems test your ability to recognize how a change in one variable (mass) affects another (kinetic energy) when a third variable (momentum) remains constant. As outlined in NCERT Class 9 Science, understanding this inverse proportionality is essential for solving mechanical physics problems without getting lost in complex calculations.

To arrive at the correct answer, identify what is fixed and what is changing. Since both bodies have the equal linear momentum (p), the kinetic energy is solely determined by the mass in the denominator. By setting up the ratio of K_A to K_B, you are essentially comparing 1/m to 1/4m. When you divide these fractions, the m cancels out and the 4 moves to the numerator, resulting in a 4 : 1 ratio. This confirms the physical intuition that if two objects have the same momentum, the lighter body must be moving significantly faster, thus carrying more kinetic energy. Therefore, (B) 4 : 1 is the correct choice.

UPSC frequently uses distractors like Option (A) 1 : 4 to catch students who correctly identify the numbers but mistake direct proportionality for inverse proportionality. Option (C) 1 : 1 is a classic trap for candidates who assume that equality in one motion vector (momentum) implies equality in all others. Always perform a sanity check: because body A is lighter (m) than body B (4m), its energy must be higher. This simple logic immediately eliminates options where the first number is smaller than or equal to the second, leaving (B) as the only mathematically sound answer.