A ball is thrown vertically in vacuum. It goes up and falls down on Earth. Which of the given figures represents its true time velocity graph?

1. Scalars, Vectors, and Kinematic Variables (basic)

To understand how things move, we must first distinguish between two types of physical quantities: Scalars and Vectors. A scalar quantity is defined entirely by its magnitude (size), such as the distance between Delhi and Mumbai (1419 km) Geography of India by Majid Husain, Transport, Communications and Trade, p.3. In contrast, a vector quantity requires both magnitude and a specific direction. For instance, while we might say a jet stream has a speed of 120 kmph, its velocity is a vector because it describes that speed moving in a specific direction across the atmosphere Physical Geography by PMF IAS, Jet streams, p.386.

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. Within this motion, we track several key variables:

• Distance vs. Displacement: Distance is the total path covered (scalar), while displacement is the straight-line change in position from start to finish (vector).
• Speed vs. Velocity: Speed is the rate of covering distance. Velocity is the rate of displacement. If a train moves at a constant speed in one direction, it is in uniform linear motion Science-Class VII NCERT (Revised ed 2025), Measurement of Time and Motion, p.117.
• Acceleration: This is the rate at which velocity changes. If a train slows down to enter a station, its motion is non-uniform because its velocity is changing over time.

Feature	Scalar	Vector
Definition	Magnitude only	Magnitude + Direction
Examples	Distance, Speed, Time, Mass	Displacement, Velocity, Acceleration, Force
Change	Changes if magnitude changes	Changes if magnitude OR direction changes

Sources: Science-Class VII NCERT (Revised ed 2025), Measurement of Time and Motion, p.116-117; Geography of India by Majid Husain, Transport, Communications and Trade, p.3; Physical Geography by PMF IAS, Jet streams, p.386

2. Acceleration Due to Gravity (g) and Free Fall (basic)

When we drop an object, it falls toward the Earth because of the Gravitational Force. If we ignore air resistance—treating the environment like a vacuum—the object is said to be in Free Fall. During this motion, the only force acting on the object is gravity, which causes it to accelerate at a constant rate known as the Acceleration Due to Gravity (g). On Earth, this value is approximately 9.8 m/s², though it is much stronger on the Sun (274 m/s²) and weaker on the Moon (1.62 m/s²) Physical Geography by PMF IAS, The Solar System, p.23.

It is important to understand that g is not perfectly uniform across the entire Earth. Two main factors influence its local value:

• Shape of the Earth: Because the Earth is slightly flattened at the poles and bulges at the equator, the distance to the center is shorter at the poles. Consequently, gravity is greater at the poles and less at the equator FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19.
• Mass Distribution: The mass within the Earth's crust is not spread evenly. Areas with higher mass density exert a stronger pull. The difference between the observed gravity and the expected value is called a Gravity Anomaly Physical Geography by PMF IAS, Earths Interior, p.58.

When an object is thrown vertically upward, gravity acts on it continuously in the downward direction. As the object rises, gravity slows it down (negative acceleration) until its velocity reaches zero at the highest point. Then, as it falls back down, gravity speeds it up in the negative (downward) direction. Throughout this entire flight—upward, at the peak, and downward—the acceleration remains constant at g. If you were to plot this on a velocity-time graph, it would appear as a single straight line with a constant negative slope, crossing the time axis at the moment the object reaches its peak height.

Phase of Motion	Velocity (v)	Acceleration (g)
Rising Upward	Decreasing (Positive)	Constant 9.8 m/s² (Downward)
At the Peak	Zero	Constant 9.8 m/s² (Downward)
Falling Downward	Increasing (Negative)	Constant 9.8 m/s² (Downward)

Sources: Physical Geography by PMF IAS, The Solar System, p.23; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58

3. Equations of Motion for Uniform Acceleration (intermediate)

In mechanics, Uniform Acceleration refers to motion where the velocity of an object changes by the same amount in every equal interval of time. This means the rate of change of velocity—acceleration—remains constant. To describe this motion precisely, we use three fundamental Equations of Motion. If an object starts with an initial velocity (u), reaches a final velocity (v) over a time (t) with a constant acceleration (a), and covers a displacement (s), the relationships are:
1. v = u + at (Velocity-Time relation)
2. s = ut + ½at² (Position-Time relation)
3. v² = u² + 2as (Velocity-Position relation)

The most common natural example of uniform acceleration is an object in free fall near the Earth's surface. Here, the acceleration is the constant gravitational force, denoted as 'g' (approximately 9.8 m/s² downward). As noted in Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267, gravity is a fundamental force that dictates the motion of all bodies. When we apply these equations to a ball thrown vertically upward, we treat the upward direction as positive and gravity as a negative acceleration (-g). Because the acceleration is constant, the Velocity-Time (v-t) graph for this motion is always a straight line.

Imagine throwing a ball straight up: initially, it has a high positive velocity. As it rises, gravity slows it down at a constant rate until its velocity hits zero at the highest point. Then, it begins to fall, and its velocity becomes increasingly negative (downward). This entire journey—from the moment it leaves your hand until it returns—is represented by a single continuous straight line with a negative slope on a v-t graph. This illustrates that the acceleration due to gravity is acting uniformly throughout the flight, even at the very top where the ball momentarily stops NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p. 72.

Sources: Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267; NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.72

4. Air Resistance, Drag, and Terminal Velocity (intermediate)

In our previous discussions, we looked at how gravity accelerates objects downward. However, in the real world, objects don't move through a vacuum; they move through fluids like air or water. When an object moves through air, it experiences a resistive force known as Air Resistance or Drag. This force acts in the opposite direction to the motion. Unlike the constant force of gravity, drag is dynamic—it increases as the speed of the object increases. This is why high-speed phenomena, like jet streams in the upper troposphere, are influenced heavily by air density; where the air is less dense, friction is lower, allowing for much higher velocities (Physical Geography PMF IAS, Jet streams, p. 386).

To understand Terminal Velocity, imagine a skydiver jumping from a plane. Initially, the only significant force is gravity, so the skydiver accelerates downward at 9.8 m/s². As their speed increases, the upward air resistance also increases. Eventually, a point is reached where the upward drag force exactly equals the downward gravitational force. According to Newton's Second Law (F = ma), when the net force is zero, the acceleration becomes zero. The object stops speeding up and continues to fall at a constant maximum velocity. This state of equilibrium is similar to how forces balance when objects float or sink in water (Science Class VIII, Exploring Forces, p. 76).

Feature	Motion in a Vacuum	Motion in Air (with Drag)
Acceleration	Constant (g)	Decreases as speed increases
Final Velocity	Continues to increase	Becomes constant (Terminal Velocity)
Forces	Only Gravity	Gravity vs. Drag

Several factors determine how quickly an object reaches terminal velocity. Surface area is crucial; a flat sheet of paper reaches terminal velocity almost instantly because it hits many air molecules, whereas a crumpled ball of the same paper falls much faster. Furthermore, fluid density plays a role—objects fall differently in water than in air because the upward buoyant and resistive forces are much stronger in denser mediums (Science Class VIII, Exploring Forces, p. 76).

Sources: Physical Geography by PMF IAS, Jet streams, p.386; Science Class VIII NCERT, Exploring Forces, p.76

5. Basics of Projectile Motion (intermediate)

When we explore the mechanics of an object thrown vertically upward, we are observing a classic case of rectilinear motion under the constant influence of gravity. Unlike horizontal motion which might be uniform, vertical motion is non-uniform linear motion because the object's speed is constantly changing due to the Earth's gravitational pull Science-Class VII, Measurement of Time and Motion, p.117.

The journey of a projectile thrown straight up can be broken down into three distinct phases:

• The Ascent: As the object moves upward, gravity acts in the opposite direction (downward). This causes the object to slow down. In physics terms, its positive velocity is decreasing.
• The Peak: At the highest point, the object stops momentarily. Here, its velocity is exactly zero Science, Class VIII, Exploring Forces, p.72. However, it is crucial to remember that acceleration due to gravity (g) is still acting on it—otherwise, it would just float there!
• The Descent: As it falls back down, gravity and the direction of motion are now aligned. The object speeds up, and its velocity becomes increasingly negative (assuming we define "up" as the positive direction).

Mathematically, if we ignore air resistance, the acceleration is constant throughout the entire flight. This means the velocity-time graph for this motion is a single, continuous straight line with a constant negative slope. It starts high (initial upward throw), crosses the time axis (the stop at the peak), and continues downward into negative values (falling back down).

Phase of Motion	Direction of Velocity	Direction of Acceleration	Speed Trend
Going Up	Upward (+)	Downward (-)	Decreasing
At the Top	Zero	Downward (-)	Momentary Stop
Coming Down	Downward (-)	Downward (-)	Increasing

Sources: Science, Class VIII, Exploring Forces, p.72; Science-Class VII, Measurement of Time and Motion, p.117

6. Sign Conventions in Vertical Motion (exam-level)

In physics, to solve any problem involving motion, we must first establish a coordinate system. Just as we use the New Cartesian Sign Convention in optics to measure distances from the pole of a mirror Science, Class X, Light – Reflection and Refraction, p.142, we apply a similar logic to vertical motion. By standard convention, we treat the point of projection as the origin (0,0). Any vector pointing upward (like initial velocity when throwing a ball) is assigned a positive (+) sign, while any vector pointing downward (like the pull of gravity) is assigned a negative (–) sign.

Crucially, the acceleration due to gravity (g) always acts toward the center of the Earth Science, Class VIII, Exploring Forces, p.72. Therefore, in our standard "up is positive" system, a = –g (approximately –9.8 m/s²) throughout the entire flight. This doesn't change when the object starts falling; gravity doesn't "switch directions" just because the ball does. It is a constant downward tug. This is why an object thrown upward slows down: its velocity is positive but its acceleration is negative, acting as a brake until the velocity reaches zero at the highest point Science, Class VIII, Exploring Forces, p.78.

Phase of Motion	Velocity (v) Sign	Acceleration (a) Sign	Physical Behavior
Moving Upward	Positive (+)	Negative (–)	Slowing down (Deceleration)
At the Top	Zero (0)	Negative (–)	Momentary stop before turning
Moving Downward	Negative (–)	Negative (–)	Speeding up (Acceleration)

When we plot this on a velocity-time graph, the motion appears as a single continuous straight line with a constant negative slope. The line starts high in the positive region (initial upward throw), crosses the x-axis (the peak where v = 0), and continues straight down into the negative region (falling back down). The slope of this line represents the constant acceleration, –g.

Sources: Science, Class X, Light – Reflection and Refraction, p.142; Science, Class VIII, Exploring Forces, p.72; Science, Class VIII, Exploring Forces, p.78

7. Interpreting Velocity-Time (v-t) Graphs (exam-level)

To master mechanics, you must treat a Velocity-Time (v-t) graph as a visual story of an object's motion. The vertical axis represents how fast the object is going and in what direction, while the horizontal axis tracks time. The most vital rule to remember is that the slope of a v-t graph represents acceleration. If the graph is a straight line, the acceleration is constant, meaning the velocity is changing by equal amounts in equal time intervals Science-Class VII, Measurement of Time and Motion, p.117. Conversely, a curved line would indicate non-uniform acceleration.
When we analyze a ball thrown vertically upward, we are looking at uniform acceleration due to gravity (g). In a vacuum, gravity acts as a constant downward force, pulling the object toward the Earth at approximately 9.8 m/s². Because this acceleration is constant and downward, the graph must be a single straight line with a constant negative slope. Initially, the ball has a high positive velocity (moving up). As gravity slows it down, the line approaches the time axis. At the very peak of its flight, the velocity is zero—this is the point where the graph crosses the x-axis.
After reaching the peak, the ball begins to fall. Its velocity now becomes negative because it is moving in the opposite direction (downward). However, because the force of gravity hasn't changed, the slope remains identical. The graph continues as the same straight line into the negative region. A common mistake is to think the graph should 'bounce' back up like a 'V' shape, but that would imply the acceleration suddenly changed direction, which doesn't happen. Much like how a linear V–I graph shows a constant ratio of resistance Science, class X, Electricity, p.176, this linear v-t graph shows a constant ratio of acceleration.

Graph Feature	Physical Meaning
Slope	Acceleration (Steeper = higher acceleration)
Area under curve	Displacement (Total distance with direction)
Horizontal Line	Zero acceleration (Constant velocity)
Crossing X-axis	Momentary rest / Change in direction

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Science , class X (NCERT 2025 ed.), Electricity, p.176

8. Solving the Original PYQ (exam-level)

Now that you have mastered Newtonian Kinematics and the behavior of acceleration due to gravity (g), this question brings those building blocks together. In a vacuum, we eliminate air resistance, meaning the only force acting on the ball is gravity, which provides a constant downward acceleration. Because velocity is a vector quantity, we must account for direction: upward is defined as positive and downward as negative. According to the first equation of motion, v = u - gt, the relationship between time and velocity must be a linear function with a constant negative slope.

To arrive at the correct answer, simply track the ball's journey through time: at the moment of release, it has a maximum positive velocity. As it rises, gravity reduces this velocity at a steady rate until it hits zero at the highest point. As it begins its descent, the velocity becomes increasingly negative. This continuous, uniform change is represented by a single straight line that starts high on the y-axis, crosses the x-axis, and continues downward. Therefore, Figure II is the only representation that correctly mirrors this physical reality. As noted in NCERT Science Class VIII, a constant force like gravity results in a uniform change in velocity.

UPSC often uses common misconceptions as distractors to test your depth of understanding. For instance, a curved line (as seen in some options) would incorrectly suggest that acceleration is changing over time, which contradicts the law of uniform acceleration in a vacuum. Other graphs might show the velocity returning to a positive value after hitting zero; this is a classic trap that represents speed (a scalar) rather than velocity (a vector). By recognizing that the slope must remain constant and negative throughout the entire flight, you can confidently choose (B) II and avoid these conceptual pitfalls.