A ball is dropped from the top of a high building with a constant acceleration of 9.8 m/s2. What will be its velocity after 3 seconds ?

1. Scalars and Vectors in Motion (basic)

Welcome to your first step in mastering mechanics! To understand how objects move, we must first distinguish between two types of physical quantities: Scalars and Vectors. At its simplest, a Scalar quantity is defined only by its magnitude (a numerical value and a unit). It answers the question "how much?" without caring about the direction. For example, when we calculate the distance a vehicle covers—say, 2 km—we are using a scalar quantity Science-Class VII, Measurement of Time and Motion, p.119. Other common scalars include mass, temperature, and time.

In contrast, a Vector quantity requires both magnitude and direction to be fully described. It tells us not just "how much," but also "where to." A classic example is velocity. While speed tells us a Jet Stream moves at 120 kmph, its velocity specifies that it flows in a particular direction, such as west to east Physical Geography by PMF IAS, Jet streams, p.386. If you change either the speed or the direction, the vector changes. This distinction is vital because in physics, a car turning a corner at a constant speed is still "accelerating" because its direction (and thus its velocity vector) is changing.

Understanding these is essential even in geography. For instance, when we measure the latitudinal and longitudinal extent of India, we are looking at distances in specific directions across the Earth's grid INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2. To help you distinguish them at a glance, refer to the table below:

Feature	Scalar	Vector
Definition	Only Magnitude	Magnitude + Direction
Examples	Distance, Speed, Time, Mass	Displacement, Velocity, Acceleration, Force
Changes when...	Only magnitude changes	Magnitude OR Direction changes

Sources: Science-Class VII, Measurement of Time and Motion, p.119; Physical Geography by PMF IAS, Jet streams, p.386; INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2

2. Newton’s Laws of Motion (intermediate)

To understand why things move the way they do, we must look at Newton’s Laws of Motion, which represent the climax of the scientific revolution Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. These three laws provide a bridge between the cause of motion (Force) and the effect (Acceleration). The First Law (Inertia) tells us that objects are stubborn; they will maintain their state of rest or uniform motion unless an external force intervenes. Think of it as the "status quo" law of physics.

The Second Law is where the math happens: F = ma. It states that the force acting on an object is equal to its mass times its acceleration. This is crucial for understanding gravity. When an object falls, the "force" is its weight, which pulls it downward, causing its speed to increase steadily Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.72. Because Earth exerts a gravitational force on every object, a released ball doesn't just hang in the air; it accelerates at a constant rate (g ≈ 9.8 m/s²), changing its velocity over time.

Finally, the Third Law reminds us that action and reaction are equal and opposite. If the Earth pulls a ball down, the ball technically pulls the Earth up with the same force! However, because the Earth’s mass is so enormous, its acceleration is undetectable. This interconnectedness of forces is what keeps planets in their orbits and allows us to calculate the precise speed of an object after it has been falling for a specific duration.

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.72

3. Understanding Acceleration (basic)

In our previous steps, we looked at speed and motion. However, in the real world, objects rarely move at a perfectly steady pace. As we observe in daily life, most motion is non-uniform, meaning the speed of an object keeps changing as it moves Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. This change in motion is captured by the concept of acceleration.

Acceleration is defined as the rate of change of velocity per unit of time. While speed tells you how fast you are going, acceleration tells you how quickly your speed (or direction) is changing. Mathematically, we express this relationship using the first equation of motion: v = u + at. In this formula, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time elapsed. If an object starts from a stationary position, or "rest," its initial velocity (u) is zero.

One of the most important examples of constant acceleration is gravity. When an object is dropped from a height, it doesn't fall at a constant speed; it actually speeds up as it falls. On Earth, this gravitational acceleration (denoted as g) is approximately 9.8 m/s². This means that for every second an object falls, its downward speed increases by exactly 9.8 meters per second. This is why a ball dropped from a building will be moving significantly faster after three seconds than it was after just one second.

It is also vital to remember that acceleration is a vector quantity, meaning it has a direction. In geography, we see this when air flows around centers of atmospheric pressure. Even if the wind speed remains constant, the change in its direction as it circles a cyclone is a form of centripetal acceleration Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309. Whether you are speeding up a car or a wind current is curving into a vortex, acceleration is the driving force behind the change.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

4. Universal Law of Gravitation (intermediate)

The Universal Law of Gravitation, formulated by Isaac Newton, was the climax of the scientific revolution, unifying the heavens and the Earth under a single physical law Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. It states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is expressed as: F = G (m₁m₂) / r², where G is the Universal Gravitational Constant, m₁ and m₂ are the masses, and r is the distance between them.

A critical distinction to master for the UPSC is the difference between mass and weight. While mass is an intrinsic property of an object (the amount of matter it contains) and remains constant everywhere, weight is a force. Weight is specifically the gravitational pull exerted by a celestial body (like Earth or the Moon) on an object Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.75. Because the gravitational pull varies depending on the mass and radius of a planet, your weight would change if you traveled from Earth to Mars, even though your mass remains the same.

Feature	Mass	Weight
Nature	Scalar quantity (only magnitude).	Vector quantity (magnitude and direction).
Location	Constant throughout the universe.	Varies from place to place based on gravity.
Measurement	Measured in Kilograms (kg).	Measured in Newtons (N) using a spring balance.

In advanced physics, we see the extremes of this law. Einstein’s General Relativity refined Newton’s work by describing gravity as the curvature of spacetime. When mass becomes incredibly dense, it can create a singularity—a point where gravity is so intense that the known laws of physics cease to exist Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.7. Indian astrophysicist Subrahmanyan Chandrasekhar contributed significantly to this field by determining the maximum mass (the Chandrasekhar Limit) a star can have before it collapses into a white dwarf or a black hole.

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.74-75; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5-7

5. Circular Motion and Centripetal Force (exam-level)

To understand circular motion, we must first distinguish it from the motion we see on a straight track. In linear motion, an object moves along a straight line, and if it covers equal distances in equal intervals of time, it is in uniform motion Science-Class VII . NCERT, Measurement of Time and Motion, p.117. However, circular motion introduces a fascinating twist: even if an object moves at a constant speed, its direction is constantly changing at every single point on the circle. Because velocity includes both speed and direction, any change in direction means the velocity is changing, which mathematically implies that the object is accelerating.

This acceleration requires a force to sustain it. The Centripetal Force is the "center-seeking" force that acts at right angles to the motion, pulling the object toward the center of the rotation. In our atmosphere, this force is what creates the circular patterns of wind (vortices) around high and low-pressure systems, leading to the formation of cyclones and anticyclones Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

Conversely, from the perspective of the rotating body, there is an apparent outward force known as Centrifugal Force. While centripetal force pulls inward, centrifugal force acts outward. This balance is crucial in physical geography: the Earth's rotation creates a centrifugal force that is strongest at the equator, causing our planet to bulge outward and making gravity slightly weaker there compared to the poles Physical Geography by PMF IAS, Latitudes and Longitudes, p.241. This same interplay between the Moon's gravitational pull and centrifugal force is responsible for the tidal bulges we see in our oceans FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT), Movements of Ocean Water, p.109.

Feature	Centripetal Force	Centrifugal Force
Direction	Inward (towards the center)	Outward (away from the center)
Nature	Real force required for circular paths	Apparent/Inertial force due to rotation
Example	Wind rotating around a Low Pressure	The bulge of the Earth at the Equator

Sources: Science-Class VII . NCERT, Measurement of Time and Motion, p.117; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309; Physical Geography by PMF IAS, Latitudes and Longitudes, p.241; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT), Movements of Ocean Water, p.109

6. Work, Energy, and Conservation (exam-level)

To understand mechanics, we must first master the relationship between Work and Energy. In physics, work is done when a force acting on an object causes it to move. This work isn't 'lost' but is transformed into energy—the capacity to perform work. As noted in the Law of Conservation of Energy, energy in a closed system is neither created nor destroyed; it merely changes form Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. For instance, wind turbines do not create energy; they transform the Kinetic Energy (energy of motion) of atmospheric air into mechanical power, which is then converted into electricity Environment, Shankar IAS Academy, Renewable Energy, p.290.

When we look at objects in motion, we often use Kinematics to describe how energy transformations manifest as changes in speed. A classic example is a ball dropped from a height. Here, gravity—a non-contact force—does work on the ball Science, Class VIII, NCERT, Exploring Forces, p.69. As the ball falls, its Potential Energy (due to height) converts into Kinetic Energy. To calculate its final speed (v) after a specific time (t), we use the first equation of motion: v = v₀ + at, where 'v₀' is the initial velocity and 'a' is the acceleration. In a vacuum, all objects fall with a constant gravitational acceleration (g) of approximately 9.8 m/s².

It is important to remember that while the total energy of a system remains constant, the quality or usability of that energy often changes. When work is performed, some energy is inevitably dissipated, usually as heat Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. In biological systems, this is why energy flow is unidirectional; as energy moves up trophic levels, a significant portion is 'lost' to the environment through respiration, even though the total mass of matter in the biosphere remains balanced through cyclic pathways.

Sources: Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Environment, Shankar IAS Academy, Renewable Energy, p.290; Science, Class VIII, NCERT, Exploring Forces, p.69

7. Equations of Motion for Uniform Acceleration (intermediate)

When we study objects in motion, we often encounter scenarios where the speed is changing. If the speed changes at a constant rate, we call this Uniform Acceleration. This is a special case of non-uniform motion, which is much more common in our daily lives than perfectly steady motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119. For example, a train leaving a station accelerates from a slow speed to a faster one in a linear motion—moving along a straight path Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. To predict exactly where that train will be or how fast it will be going at any second, we use the three Equations of Motion.

These equations are the mathematical bridge between displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). They only work when acceleration is constant. Let’s look at them:

• v = u + at: The Velocity-Time relation. It tells you the final speed if you know the starting speed and how long it accelerated.
• s = ut + ½at²: The Position-Time relation. It calculates how far an object has traveled over a period of time.
• v² = u² + 2as: The Velocity-Displacement relation. This is your go-to formula when you don't know the time duration but know the distance covered.

In physics problems, a very common form of uniform acceleration is gravity. When an object is dropped, it accelerates downward at approximately 9.8 m/s². To simplify your calculations, always identify your "givens" first. If an object starts from rest, your initial velocity (u) is 0. If it comes to a stop, your final velocity (v) is 0.

Motion Type	Acceleration	Velocity
Uniform Motion	Zero	Constant
Uniformly Accelerated Motion	Constant (Non-zero)	Changes at a steady rate

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116

8. Dynamics of Free Fall (exam-level)

When we speak of **free fall**, we are describing a specific type of **non-uniform linear motion** where an object moves vertically under the sole influence of gravity Science-Class VII, Measurement of Time and Motion, p.117. In this state, the object’s speed is constantly changing. If you drop an object from rest, it doesn't just 'float' down at a steady pace; it accelerates. This acceleration is a result of the Earth's gravitational pull, which acts as the 'switch' for almost all movement on our planet’s surface FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.38. On Earth, this constant acceleration, denoted as g, is approximately 9.8 m/s² Physical Geography by PMF IAS, The Solar System, p.23. This means that for every second an object falls, its downward speed increases by 9.8 meters per second. To calculate the dynamics of this motion, we use the primary kinematic relation: v = u + at. In the context of a 'dropped' object, the initial velocity (u) is 0. The acceleration (a) is g (9.8 m/s²), and t represents the time elapsed. Therefore, the velocity after any given time is simply the product of gravity and time (v = gt). It is fascinating to note that while the object moves in a straight vertical path, its behavior changes based on direction: when thrown upward, it slows down until it stops momentarily at the peak; when falling downward, its speed increases steadily Science, Class VIII, Exploring Forces, p.72. In standard physics problems, we assume a 'vacuum' environment, meaning we neglect air resistance to focus purely on gravitational dynamics.

Feature	Upward Motion	Downward (Free Fall)
Speed	Decreases (Deceleration)	Increases (Acceleration)
Acceleration	-9.8 m/s² (opposing motion)	+9.8 m/s² (assisting motion)
Final Velocity	Zero at the highest point	Maximum just before impact

Sources: Science-Class VII, Measurement of Time and Motion, p.117; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.38; Physical Geography by PMF IAS, The Solar System, p.23; Science, Class VIII, Exploring Forces, p.72

9. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental kinematic equations, this question serves as a direct application of the first equation of motion: v = u + at. In UPSC General Science, the term "dropped" is a critical conceptual trigger signifying that the initial velocity (u) is exactly zero. By identifying the building blocks you just learned—the constant acceleration of 9.8 m/s² and a time interval of 3 seconds—you are observing how velocity accumulates linearly over time in a free-fall scenario, a principle explained in NASA's Guide to Falling Objects.

To arrive at the correct answer (C) 29.4 m/s, walk through the logic of constant change: if the speed increases by 9.8 m/s every single second, then after three seconds, the calculation is a straightforward multiplication (9.8 × 3). This step-by-step accumulation is the essence of uniform acceleration. As noted in Tennessee Tech Kinematics, neglecting air resistance allows us to use this clean linear relationship to predict motion with high precision.

UPSC designed the distractors in this question to catch calculation lapses or conceptual rushing. Options (A) 9.8 m/s and (B) 19.6 m/s are "partial progress" traps; they represent the velocity after only 1 and 2 seconds respectively. Option (D) 39.2 m/s is the velocity after 4 seconds. The examiners are testing your attention to detail—ensuring you don't just know how to calculate, but that you use the specific time variable (3 seconds) provided in the prompt. Always verify that your final value corresponds exactly to the time-stamp requested.