Two cars A and B have masses mA and mB respectively, with mA > mB- Both the cars are moving in the same direction with equal kinetic energy. If equal braking force is applied on both, then before coming to rest

1. Newton’s Laws and the Concept of Mass (basic)

To understand mechanics, we must first master the distinction between Mass and Weight, a common point of confusion. In physics, mass is an intrinsic property of an object; it represents the amount of matter it contains and, more importantly, it is a measure of Inertia. Inertia is the natural tendency of an object to resist any change in its state of rest or uniform motion. As explained in Science, Class VIII, Exploring Forces, p.75, mass remains constant everywhere in the universe, whether you are on Earth or the Moon.

Weight, on the other hand, is a force. It is the measure of the gravitational pull exerted on an object by a massive body like the Earth. Because gravitational force varies slightly depending on your location (and significantly on different planets), your weight can change while your mass stays the same Science, Class VIII, Exploring Forces, p.75. For instance, weight is measured using tools like a spring balance, which reads force in Newtons (N) rather than kilograms Science, Class VIII, Exploring Forces, p.74.

Newton’s laws connect these concepts. His Second Law (F = ma) tells us that the force (F) required to move an object depends directly on its mass (m). A larger mass requires a greater force to achieve the same acceleration. This relationship is fundamental to understanding everything from the motion of cars to the gravitational forces that govern the rotation and revolution of the Earth Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267.

Feature	Mass	Weight
Nature	Intrinsic property (Inertia)	Force of gravity
Location	Constant everywhere	Changes with gravity
Measurement	Measured in Kilograms (kg)	Measured in Newtons (N)

Sources: Science, Class VIII, Exploring Forces, p.74; Science, Class VIII, Exploring Forces, p.75; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267

2. Kinetic Energy and Its Determinants (basic)

Kinetic Energy (KE) is the energy an object possesses by virtue of being in motion. From a first-principles perspective, if an object is at rest, it has zero kinetic energy; as soon as it starts moving, it acquires energy that can perform work. This is beautifully illustrated in how we harness renewable power: for instance, wind energy is simply the conversion of the kinetic energy of blowing wind into electrical energy through the rotation of turbines INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Mineral and Energy Resources, p.61. Even at a microscopic level, what we perceive as temperature is actually a measurement of the average kinetic energy of vibrating molecules in the air Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8.

The amount of kinetic energy an object holds is determined by two primary factors: its mass (m) and its velocity (v). The mathematical relationship is expressed as KE = ½mv². Because the velocity is squared, it has a disproportionately large impact on the total energy—doubling the speed of a vehicle doesn't just double its energy; it quadruples it. This explains why high-speed collisions are significantly more damaging than low-speed ones, even for the same vehicle.

Crucially, we must understand the Work-Energy Theorem, which states that the net work done on an object is equal to its change in kinetic energy (W = ΔK). In practical terms, to stop a moving vehicle, a braking force must do enough "work" to reduce that kinetic energy to zero. Since work is the product of Force (F) and Displacement (d), the distance it takes to stop depends entirely on the initial energy and the force applied Science, class X (NCERT 2025 ed.), Electricity, p.173.

Factor	Relationship with KE	Example
Mass (m)	Linear (KE ∝ m)	A truck at 20 km/h has more KE than a cycle at 20 km/h.
Velocity (v)	Exponential (KE ∝ v²)	A car at 60 km/h has 9 times the KE it has at 20 km/h.

Sources: INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Mineral and Energy Resources, p.61; Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8; Science, class X (NCERT 2025 ed.), Electricity, p.173

3. The Concept of Mechanical Work (basic)

In common language, we use the word 'work' to describe any physical or mental effort. However, in the precise world of Mechanics, work has a very specific definition. Mechanical Work is done only when a force acts upon an object and causes it to move through a displacement. If you push against a heavy stone wall for an hour, you will feel exhausted, but because the wall has not moved, the mechanical work done is strictly zero. This distinguishes the physical concept from the socio-economic definition of 'work' which focuses on time and productivity, such as the classification of a Main Worker as someone who works for at least 183 days a year INDIA PEOPLE AND ECONOMY, Population: Distribution, Density, Growth and Composition, p.11.

Mathematically, work (W) is calculated as the product of the force (F) applied and the displacement (d) in the direction of the force: W = F × d × cosθ, where θ is the angle between the force and the displacement vector. It is a scalar quantity, meaning it has magnitude but no direction, and its SI unit is the Joule (J). Just as inputs are necessary for production in an economy Microeconomics, Production and Costs, p.38, a non-zero force and a non-zero displacement are the 'inputs' required to produce mechanical work.

The direction of the force relative to the movement is crucial. We can categorize work into three types based on the angle θ:

Type of Work	Angle (θ)	Description	Example
Positive	0° ≤ θ < 90°	Force and displacement are in the same general direction.	Kicking a ball forward.
Negative	90° < θ ≤ 180°	Force opposes the motion.	Friction acting on a sliding box.
Zero	θ = 90°	Force is perpendicular to the displacement.	A coolie carrying a load on his head while walking horizontally.

Sources: INDIA PEOPLE AND ECONOMY, Population: Distribution, Density, Growth and Composition, p.11; Microeconomics, Production and Costs, p.38

4. Law of Conservation of Energy (intermediate)

The Law of Conservation of Energy is one of the most fundamental principles in physics and mechanics. It states that energy can neither be created nor destroyed; it can only be transformed from one form to another. In any isolated system, the total amount of energy remains constant. For instance, in an ecosystem, solar energy is converted into chemical energy through photosynthesis by primary producers Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.31. While the energy flows through the food chain, it is often dissipated as heat during respiration, but the total energy in the universe remains unchanged Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14.

In the realm of mechanics, we often look at this through the Work-Energy Theorem. This theorem tells us that the net work done on an object is exactly equal to the change in its Kinetic Energy (ΔK). Work (W) is defined as the product of the force (F) applied and the displacement (d) over which it acts (W = F × d). This means if you want to change an object's energy—say, by bringing a speeding car to a halt—you must perform work on it. Because our systems are rarely 100% efficient, much of this mechanical work is converted into thermal energy (heat) due to friction, which is why India emphasizes the Energy Conservation Act of 2001 to improve efficiency and reduce waste in our national infrastructure Contemporary World Politics, Environment and Natural Resources, p.90.

A fascinating implication of this law occurs when comparing two objects of different masses. If a heavy truck and a light car possess the same initial kinetic energy and you apply the same braking force to both, they will both travel the exact same distance before stopping. Why? Because the work required to reduce their kinetic energy to zero is the same for both (W = ΔK). Since W = F × d, and both 'W' and 'F' are identical for both vehicles, the stopping distance 'd' must also be identical, regardless of their mass. This highlights that energy, rather than just mass or velocity alone, is the ultimate currency of motion.

Sources: Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.31; Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Contemporary World Politics, Environment and Natural Resources, p.90

5. Friction and Braking Mechanisms (intermediate)

To understand braking, we must first look at the microscopic world. Friction is a contact force that arises because no surface is perfectly smooth. Even a polished metal floor has microscopic 'peaks' and 'valleys' called irregularities. When two surfaces touch, these irregularities interlock and oppose any effort to move one surface over the other Science, Class VIII, NCERT, Exploring Forces, p.68. In a braking system, we intentionally use this friction to convert a vehicle's kinetic energy (energy of motion) into thermal energy (heat), thereby reducing its speed Science, Class VIII, NCERT, Exploring Forces, p.65.

The core principle governing this process is the Work-Energy Theorem. It states that the net work done on an object is equal to its change in kinetic energy (ΔK). In mathematical terms, Work (W) is the product of the Force (F) applied and the Displacement (d) or distance over which that force acts: W = F · d. When you hit the brakes, the friction force (F) does 'negative work' to bring the car's kinetic energy down to zero. Therefore, the work required to stop a car is exactly equal to the kinetic energy it possessed the moment you stepped on the pedal.

A common misconception in competitive exams is that mass alone determines stopping distance. However, if two vehicles (like a heavy truck and a light car) happen to have the exact same initial kinetic energy and are stopped using the same braking force, they will cover the exact same distance before coming to a halt. This is because since W = ΔK and W = F · d, it follows that d = ΔK / F. If ΔK and F are identical for both vehicles, the distance (d) must also be identical, regardless of their individual masses or velocities.

Friction isn't just limited to mechanical brakes; it is a universal force that influences everything from the movement of your bicycle to global weather. For instance, the irregularities of the Earth's surface resist wind movement, causing surface winds to cross pressure lines (isobars) at high angles—an effect that diminishes at higher altitudes where surface friction is absent Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307.

Sources: Science, Class VIII, NCERT, Exploring Forces, p.68; Science, Class VIII, NCERT, Exploring Forces, p.65; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307

6. The Work-Energy Theorem (exam-level)

At the heart of mechanics lies the Work-Energy Theorem, a powerful principle that bridges the gap between the forces acting on an object and its resulting motion. Simply put, the net work done on an object by all acting forces is exactly equal to the change in its kinetic energy (ΔK). Mathematically, this is expressed as W = ΔK = K_final - K_initial. This concept is a specific application of the broader law that when work is done, energy is transformed from one form to another Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14.

To understand its application, consider the act of braking a vehicle. When you apply the brakes, the frictional force does negative work on the car because the force is opposite to the direction of displacement. This work "robs" the car of its kinetic energy until it comes to a stop. A crucial insight for competitive exams is the stopping distance. If two vehicles, regardless of their individual mass or velocity, possess the same initial kinetic energy and are subjected to the same braking force, they will travel the exact same distance before stopping. This is because the work required to stop them (W = Force × distance) must equal their initial kinetic energy.

It is important to distinguish between force and energy. While a heavier car might require more force to achieve the same acceleration as a lighter car, the Work-Energy Theorem tells us that if the energy state is already defined, the work done (and thus the distance covered under a constant force) depends solely on that energy value. This principle is universal, whether we are discussing a car on a road or the movement of a charge through a potential difference, where work is also defined as the product of the "driving force" (potential) and the quantity moved Science, class X (NCERT 2025 ed.), Electricity, p.173.

Sources: Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Science, class X (NCERT 2025 ed.), Electricity, p.173

7. Solving the Original PYQ (exam-level)

This question is a classic application of the Work-Energy Theorem, which you just mastered in the previous module. To solve this, you must synthesize three building blocks: the definition of Kinetic Energy, the formula for Work Done (Force × Displacement), and the principle that the work done by a force is exactly equal to the change in an object's energy. Since both cars start with equal kinetic energy and come to a complete stop (final kinetic energy is zero), the total work required to stop both car A and car B is identical. By applying the logic from Principles of Mechanics, we see that if the Work ($W$) is the same and the applied braking force ($F$) is the same, then the displacement ($d$) must also be the same because $W = F \times d$.

To arrive at (C) both will cover the same distance, you must ignore the "noise" of the different masses. While car A is heavier ($m_A > m_B$), the problem statement has already "compensated" for this by telling you their kinetic energies are already equal. If they have the same energy and experience the same opposing force, they must travel the same distance before that energy is fully dissipated. Think of it as a bank account: if two people have the same amount of money and spend it at the exact same rate per mile traveled, they will both run out of money at the exact same distance, regardless of their weight or speed.

UPSC uses Options (A), (B), and (D) as conceptual traps to exploit your intuition. Most students fall for the trap of thinking about Momentum ($p = mv$), where a heavier car would indeed be harder to stop if the velocities were equal. However, the question specifies equal energy, not equal velocity. Option (D) is a classic "distractor" designed to make you think the information is insufficient, but the Work-Energy Theorem provides a complete 1:1 relationship between force, distance, and energy change, making the specific velocity values irrelevant to the final outcome.