A car moving with a speed of 12 m/s is subjected to brakes which produces a deceleration of 6 m/s². The car takes 2 s to stop after the application of brakes. What is the distance covered by the car after the application of brakes?

1. Basics of Motion: Distance vs. Displacement (basic)

Welcome to the first step of your journey into mechanics! To master how things move, we must first distinguish between the path we take and where we actually end up. Imagine you are tracking a vehicle's movement. The distance is the total length of the actual path covered by the object. It is a scalar quantity, meaning it only has a magnitude (size) and doesn't care about direction. Whether you walk in circles or a straight line, every step adds to your distance. For instance, in geography, we measure the actual distance between the north and south extremities of India as 3,214 km INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p. 2.

On the other hand, displacement is the shortest straight-line path between your starting point and your final destination. It is a vector quantity, which means it requires both a magnitude and a specific direction (e.g., "5 km North"). If you run a full lap around a park and return to your starting gate, your distance might be 1 km, but your displacement is zero because your position hasn't changed relative to the start.

Understanding this difference is crucial when we analyze motion influenced by forces. For example, when brakes are applied to a car, we measure the distance it travels before coming to a stop to understand its safety and the force of friction involved Science, Class VIII, Exploring Forces, p. 68. While distance can only stay the same or increase as you move, displacement can decrease, become zero, or even become negative depending on your direction.

Feature	Distance	Displacement
Definition	Total path length traveled.	Shortest distance between start and end.
Type	Scalar (Magnitude only).	Vector (Magnitude + Direction).
Value	Always positive or zero.	Can be positive, negative, or zero.

Sources: INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2; Science, Class VIII, Exploring Forces, p.68

2. Understanding Velocity and Speed (basic)

In our journey to master mechanics, we must first distinguish between how fast something moves and where it is going. At its simplest, Speed is the rate at which an object covers distance. If you are told a train covers 100 kilometers in 2 hours, you can calculate its speed as 50 km/h. However, in the real world, objects rarely move at a perfectly steady pace. A car might slow down for a speed breaker or accelerate on a highway; this is known as non-uniform motion Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.117. Because of these fluctuations, we often use the term Average Speed, which is simply the total distance covered divided by the total time taken Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.115.

While speed tells us the magnitude of motion, Velocity adds a crucial layer: direction. Velocity is speed in a specific direction. For example, knowing a jet stream moves at 200 km/h is helpful, but knowing it moves 200 km/h towards the East describes its velocity. Interestingly, the velocity of these high-altitude jet streams is heavily influenced by the temperature contrast between air masses; a sharper temperature difference leads to higher velocity Physical Geography by PMF IAS, Jet streams, p.385.

When an object moves along a straight path at a constant speed, it is in uniform linear motion. In this specific case, the object covers equal distances in equal intervals of time Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.117. If either the speed or the direction changes, the velocity changes, and the motion becomes non-uniform.

Feature	Speed	Velocity
Definition	Distance covered per unit time.	Displacement (distance in a direction) per unit time.
Components	Magnitude (Size) only.	Magnitude AND Direction.
Change	Changes only if the pace changes.	Changes if the pace OR the direction changes.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Physical Geography by PMF IAS, Jet streams, p.385

3. Acceleration and Retardation (intermediate)

In mechanics, acceleration is the rate at which an object's velocity changes over time. It is not just about how fast you are going, but how quickly your speed or direction is shifting. Because acceleration involves both magnitude and direction, it is a vector quantity. When a vehicle speeds up, the acceleration acts in the direction of motion; however, when we apply brakes to slow down, we encounter retardation (also known as deceleration). In this case, the acceleration acts in the opposite direction to the motion, which we mathematically represent with a negative sign Science-Class VII, Chapter 8, p. 113.

To calculate the motion of objects undergoing constant acceleration or retardation, we use the equations of motion. These formulas link initial velocity (u), final velocity (v), acceleration (a), time (t), and distance (s). For instance, if you need to find how much ground a car covers while braking, you can use the formula s = ut + ½at² or v² = u² + 2as. It is vital to remember that during retardation, the value of 'a' must be substituted as a negative number to reflect the reduction in speed over time.

Feature	Acceleration	Retardation (Deceleration)
Velocity Change	Velocity is increasing	Velocity is decreasing
Direction	Same as the direction of motion	Opposite to the direction of motion
Mathematical Sign	Positive (+)	Negative (-)

Beyond straight-line motion, acceleration can also occur when only the direction changes, even if the speed stays constant. A prime example is centripetal acceleration, which acts on air flowing around centers of atmospheric circulation, creating the circular patterns we see in cyclones and anticyclones Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

4. Newton’s Laws of Motion and Force (exam-level)

To understand how things move, we must look at the Laws of Motion formulated by Sir Isaac Newton. These laws provide the foundation for classical mechanics by explaining the relationship between a physical object and the forces acting upon it. At its simplest, a force is a push or a pull. In the scientific world, we measure force in newtons (N) Science, Class VIII, Chapter 6, p.65. Whether you are pushing a heavy crate or Earth is pulling you down via gravity, you are dealing with force.

Newton’s Three Laws can be summarized as follows:

• First Law (Inertia): An object will not change its motion unless a force acts on it. If it is at rest, it stays at rest; if it is moving, it keeps moving at the same speed and in the same direction.
• Second Law (F = ma): The force on an object is equal to its mass times its acceleration. This tells us that a larger force is needed to accelerate a heavier object, and that applying force results in a change in velocity (acceleration or deceleration).
• Third Law (Action/Reaction): For every action, there is an equal and opposite reaction. If you push against a wall, the wall pushes back on you with the same intensity.

It is vital to distinguish between mass and weight, as these are often confused in daily speech. While mass is the amount of matter in an object and remains constant regardless of location, weight is actually a force—specifically, the force with which the Earth pulls an object toward itself Science, Class VIII, Chapter 6, p.72. Because weight is a force, its SI unit is also the newton (N), not the kilogram Science, Class VIII, Chapter 6, p.77.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Gravitational force acting on an object.
Constancy	Remains unchanged everywhere.	Varies from place to place (e.g., Earth vs. Moon).
SI Unit	Kilogram (kg)	Newton (N)

Finally, we categorize forces based on how they interact with objects. Contact forces require physical touch, such as friction or the force applied by a stick Science, Class VIII, Chapter 6, p.66. In contrast, non-contact forces like gravity or magnetism act over a distance. Understanding these interactions allows us to calculate precisely how objects—from cars applying brakes to planets orbiting stars—will behave under different conditions.

Sources: Science, Class VIII, Exploring Forces, p.65; Science, Class VIII, Exploring Forces, p.66; Science, Class VIII, Exploring Forces, p.72; Science, Class VIII, Exploring Forces, p.77

5. Friction and Braking Systems (intermediate)

At its core, friction is a resistance force that occurs when two surfaces interact. It is classified as a contact force because it arises from the physical interlocking of microscopic irregularities on the surfaces in contact Science, Class VIII, Exploring Forces, p.68. Even a surface that looks perfectly smooth to the naked eye contains minute bumps and valleys. When one object tries to move over another, these irregularities "lock" into each other, creating a force that opposes the direction of motion Science, Class VIII, Exploring Forces, p.77.

In braking systems, we intentionally harness this force to control or stop motion. When you press the brake pedal in a car, the system forces brake pads against a rotating disc or drum. This creates immense friction, which converts the vehicle's kinetic energy (energy of motion) into heat energy. This is similar to how the sinking of heavy elements in the Earth's early history generated heat through friction Physical Geography PMF IAS, Earths Interior, p.59. The resulting force acts in the opposite direction of the car's velocity, causing deceleration (negative acceleration).

To understand the mechanics of a stopping vehicle, we look at the relationship between speed, time, and deceleration. If a vehicle is moving at an initial speed (u) and a constant decelerating force is applied, the stopping distance (s) can be determined using kinematic equations. For instance, using the formula v² = u² + 2as (where v is the final velocity, which is 0 when stopped), we can see that the distance required to stop depends heavily on the initial speed and the rate of deceleration provided by the friction Science-Class VII, Measurement of Time and Motion, p.113.

Factor	Effect on Braking
Surface Roughness	Rougher surfaces increase friction, reducing stopping distance.
Initial Speed	Higher speeds require significantly more distance to stop.
Surface Condition	Water or oil acts as a lubricant, reducing friction and increasing stopping distance.

Sources: Science, Class VIII, Exploring Forces, p.68, 77; Physical Geography PMF IAS, Earths Interior, p.59; Science-Class VII, Measurement of Time and Motion, p.113

6. The Three Equations of Motion (exam-level)

In our previous steps, we explored how objects move and speed up. Now, we translate those concepts into the language of mathematics using the Three Equations of Motion. These equations are the bedrock of classical mechanics, allowing us to predict the future position or velocity of an object, provided it is moving in a straight line with constant acceleration.

When an object moves along a straight path, we call it linear motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. If the speed changes—such as a train starting from a station or a car applying brakes—the motion is non-uniform Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. To solve problems involving this change, we use five variables: initial velocity (u), final velocity (v), acceleration (a), time (t), and distance or displacement (s).

The three fundamental equations are:

Equation	Formula	What it relates
First Equation	v = u + at	Velocity and Time
Second Equation	s = ut + ½at²	Position (Distance) and Time
Third Equation	v² = u² + 2as	Position and Velocity

A critical point for the UPSC aspirant is the sign of acceleration. If an object is speeding up, a is positive. If it is slowing down (deceleration or retardation), a must be entered as a negative value in these formulas. For example, if a car traveling at 12 m/s applies brakes and stops (v = 0), the acceleration acting against the motion results in a distance covered that can be calculated using either the second or third equation, yielding a precise result of how far the car skids before halting.

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. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of kinematics, you can see how identifying the given variables—initial velocity (u), time (t), and deceleration (a)—is the critical first step. In this problem, the car isn't just moving; it is changing its state of motion under a constant negative acceleration. By connecting the concepts of displacement and uniformly accelerated motion, we can apply the second equation of motion, s = ut + ½at², which bridges the gap between where the car starts and where it finally rests.

Walking through the logic, we plug in our known values: u = 12 m/s, t = 2 s, and a = -6 m/s². It is vital to remember that deceleration must be treated as a negative value because it acts in the opposite direction of the motion. Calculating the first part, 12 × 2 gives us 24 meters, but we must then account for the braking force. The term ½ × -6 × 2² simplifies to -12. Thus, 24 - 12 results in 12 m, which is the correct answer (A). You could also verify this using the third equation of motion, v² = u² + 2as, where final velocity (v) is 0, confirming that your conceptual understanding is mathematically sound across different formulas as described in Science-Class VII . NCERT(Revised ed 2025).

UPSC often includes distractor options to catch common calculation errors. For instance, option (B) 24 m is a classic trap where a student might simply multiply speed by time (u × t) while forgetting that the car is slowing down. Option (C) 36 m might occur if you incorrectly add the deceleration distance instead of subtracting it, while (D) 48 m usually results from doubling the initial calculation. Always pay close attention to the negative sign in deceleration; a simple sign error is the most frequent reason candidates miss these straightforward marks.