A car accelerates from rest with acceleration 1.2 m/s2. A bus moves with constant speed of 12 m/s in a parallel lane. How long does the car take from its start to meet the bus?

1. Scalars, Vectors, and the Basics of Motion (basic)

To master mechanics, we must first understand how to describe the physical world around us. In physics, every measurable quantity falls into one of two categories: Scalars or Vectors. A Scalar quantity is defined solely by its magnitude (size). Examples include mass, time, and temperature. For instance, if you say a car is moving at 12 m/s, you are describing its speed, which is a scalar Science-Class VII, NCERT(Revised ed 2025), Chapter 8, p. 113. However, if you specify that the car is moving 12 m/s due North, you are now describing its velocity. Velocity is a Vector quantity because it requires both magnitude and a specific direction to be fully understood. Moving into the basics of motion, we often distinguish between Distance and Displacement. Distance is the total length of the path an object travels (a scalar). In contrast, Displacement is the straight-line distance between the starting and ending points, including the direction (a vector). This distinction is vital in geography as well; for example, while the latitudinal extent of India is roughly 30 degrees, the actual physical distance from North to South is 3,214 km INDIA PHYSICAL ENVIRONMENT, Geography Class XI (NCERT 2025 ed.), Chapter 1, p. 2. In physics, if a person walks 5 km East and then 5 km West, their total distance is 10 km, but their displacement is zero because they ended exactly where they started. Finally, we must understand how motion changes. Acceleration is defined as the rate of change of velocity. If an object is moving at a constant speed in a straight line, we call this uniform linear motion. If the speed or direction changes, the motion becomes non-uniform Science-Class VII, NCERT(Revised ed 2025), Chapter 8, p. 117. These concepts allow us to use kinematic equations like s = ut + 0.5at², where 's' represents displacement, 'u' is initial velocity, 'a' is acceleration, and 't' is time. Understanding these fundamentals is the first step toward calculating how objects like cars or buses interact on the road.

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

Sources: Science-Class VII, NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113, 117; INDIA PHYSICAL ENVIRONMENT, Geography Class XI (NCERT 2025 ed.), Chapter 1: India — Location, p.2

2. Uniform and Non-Uniform Motion (basic)

In our study of mechanics, we classify how objects move along a straight path based on their consistency. This brings us to the fundamental distinction between Uniform and Non-Uniform Linear Motion. To understand this, we look at whether an object's speed remains steady or fluctuates as time passes.

Uniform Linear Motion occurs when an object moves along a straight line at a constant speed Science-Class VII, Chapter 8: Measurement of Time and Motion, p.118. This means the object covers equal distances in equal intervals of time, no matter how small those intervals are. For example, if a train travels exactly 2 kilometers every minute without speeding up or slowing down, it is in uniform motion. In this state, the object’s actual speed at any moment is identical to its average speed.

In contrast, Non-Uniform Linear Motion describes a situation where the speed of an object keeps changing as it moves along a straight line Science-Class VII, Chapter 8: Measurement of Time and Motion, p.118. Most real-world movements are non-uniform. Think of a car in city traffic: it accelerates from a stoplight, slows down for a turn, and stops for pedestrians. Because the speed is inconsistent, we rely on the concept of average speed—the total distance covered divided by the total time taken—to describe the overall journey Science-Class VII, Chapter 8: Measurement of Time and Motion, p.118.

Feature	Uniform Motion	Non-Uniform Motion
Speed	Remains constant/fixed	Changes (increases or decreases)
Distance	Equal distances in equal time	Unequal distances in equal time

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118

3. The Three Kinematic Equations of Motion (intermediate)

In our journey through mechanics, the Three Kinematic Equations of Motion serve as the essential toolkit for predicting the behavior of any object moving in a straight line. However, there is a golden rule you must remember: these equations only work when acceleration (a) is constant. If acceleration changes, these formulas fall apart. As noted in foundational physics studies, understanding the distinction between uniform and non-uniform motion is the first step toward mastering these calculations Science-Class VII, NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 117.

Let’s break down the three fundamental equations and what they actually tell us about the world:

• First Equation (Velocity-Time): v = u + at
  This equation tells us the final velocity (v) of an object if we know its starting speed (u), how hard it is accelerating (a), and for how long (t). It describes how speed builds up over time.
• Second Equation (Position-Time): s = ut + ½at²
  This is the most versatile equation for solving "interception" problems. It calculates displacement (s). If a vehicle starts from rest (u = 0), the equation simplifies to s = ½at², showing that the distance covered grows with the square of time. This is why a car accelerating from a stoplight eventually catches a bus moving at a constant speed.
• Third Equation (Velocity-Position): v² = u² + 2as
  This is your go-to formula when you don't know the time (t). It relates the speed of the object directly to the distance it has traveled.

When solving complex problems, such as a faster car catching up to a slower bus, we use these equations to set up a "mathematical meeting point." By equating the displacement (s) of both vehicles, we can solve for the exact moment (t) they cross paths Science-Class VII, NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 113.

Sources: Science-Class VII, NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII, NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117

4. Motion Under Gravity: Free Fall and Projectiles (intermediate)

When we talk about Motion Under Gravity, we are exploring one of the most fundamental forces of nature. Gravity is the invisible "switch" that initiates movement for all surface material on Earth, driving everything from the flow of rivers to the fall of an apple Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.38. In mechanics, we specifically look at two scenarios: Free Fall (vertical motion) and Projectile Motion (two-dimensional motion).

Free Fall occurs when an object moves solely under the influence of gravity. When you throw a ball upwards, it doesn't move at a constant speed. Instead, it slows down as it rises, stops momentarily at its peak, and then accelerates as it falls back down Science Class VIII (NCERT 2025 ed.), Exploring Forces, p.72. This change in speed is caused by a constant acceleration known as g. On Earth, the average value of g is 9.8 m/s², though it is not the same everywhere. Because the Earth is not a perfect sphere, gravity is stronger at the poles (closer to the center) and slightly weaker at the equator Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19.

Celestial Body	Surface Gravity (m/s²)	Comparison to Earth
Sun	274 m/s²	~28 times stronger
Earth	9.8 m/s²	Standard (1g)
Moon	1.62 m/s²	~1/6th of Earth

Source: Physical Geography by PMF IAS, The Solar System, p.23

In Projectile Motion, an object is thrown with an initial velocity at an angle. To master this, you must treat the motion as two independent parts: horizontal and vertical. Horizontally, the object moves at a constant velocity (ignoring air resistance). Vertically, it is in free fall, constantly accelerating downwards at 9.8 m/s². The combination of these two paths creates the characteristic parabola curve we see when a cricketer hits a six or a hose sprays water.

Sources: Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.38; Science Class VIII (NCERT 2025 ed.), Exploring Forces, p.72; Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, The Solar System, p.23

5. Relative Velocity and Intercept Problems (intermediate)

In the study of motion, we often encounter situations where two objects are moving simultaneously. To understand when or where they will meet, we use the concept of relative velocity and intercept logic. At its core, an intercept occurs when the positions (displacements) of two objects from a common reference point become identical at the same moment in time. As we know from basic principles, the distance covered by an object is determined by its speed and the time it has been moving Science-Class VII, Chapter 8, p.115.

When solving these problems, we first define the motion of each object independently using kinematic equations. For an object moving at a constant speed, the displacement (s) is simply: s = v × t. However, if an object is accelerating from rest, its displacement grows quadratically over time, following the formula: s = ½at². By setting the displacement equations of both objects equal to each other (s₁ = s₂), we can solve for the specific time (t) at which the "catch-up" occurs.

To visualize this, consider a comparison between a steady mover and an accelerating mover:

Feature	Constant Speed Object	Accelerating Object (from rest)
Motion Type	Uniform Motion	Non-uniform Motion
Displacement Formula	s = vt	s = ½at²
Velocity Trend	Stays the same	Increases over time

Determining which object is "faster" depends on the distance covered in a specific unit of time Science-Class VII, Chapter 8, p.113. In an intercept problem, the accelerating object may start slower but will eventually overtake the constant-speed object because its velocity continues to climb. This mathematical intersection is the foundation for everything from traffic safety calculations to complex navigation in logistics Certificate Physical and Human Geography, World Communications, p.304.

Sources: Science-Class VII, Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII, Chapter 8: Measurement of Time and Motion, p.115; Certificate Physical and Human Geography, World Communications, p.304

6. Mathematical Conditions for Two Objects Meeting (exam-level)

When we talk about two objects "meeting" or one "catching up" to another in physics, we are essentially looking for a specific point in space and a moment in time where their positions are identical. To solve these problems, we rely on the fundamental principle of equating displacements. If Object A and Object B start from the same reference point, they meet when their displacement ($s$) from that starting point is exactly the same at the same time ($t$).

The mathematical approach involves writing a motion equation for each object based on its specific behavior. For an object moving at a constant speed ($v$), the displacement is simply $s = vt$. However, for an object undergoing uniform acceleration ($a$) starting from rest ($u = 0$), the displacement follows the kinematic formula $s = ½at²$ Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117. By setting these two expressions equal to each other ($s_{object1} = s_{object2}$), we create a single equation where time ($t$) is the only unknown variable.

This technique is a classic example of solving simultaneous equations by substitution. As noted in theoretical exercises, when we want to find the values of two variables (like position and time), we can solve for one in terms of the other to obtain a complete solution Macroeconomics (NCERT class XII 2025 ed.), Chapter 4: Determination of Income and Employment, p.53. In intercept problems, we usually solve for $t$ first. Once we find the time it takes to meet, we can plug that value back into either displacement equation to find the exact distance from the starting point where the encounter occurs.

Sources: Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117; Macroeconomics (NCERT class XII 2025 ed.), Chapter 4: Determination of Income and Employment, p.53

7. Solving the Original PYQ (exam-level)

This problem beautifully integrates the concepts of Uniform Motion and Uniformly Accelerated Motion that we have just explored. In competitive exams like the UPSC, "meeting" or "overtaking" problems require you to recognize a fundamental physical truth: when two objects start from the same point and meet again, their total displacements must be equal. As explained in Science-Class VII . NCERT(Revised ed 2025), the bus represents linear motion at a constant rate, while the car represents non-uniform motion where the velocity increases over time.

To arrive at the correct answer, let us walk through the logic. First, express the distance of the bus as speed × time, giving us $12t$. Next, apply the second equation of motion ($s = ut + ½at²$) for the car. Since the car starts from rest ($u = 0$), its displacement is $0.5 × 1.2 × t²$, which simplifies to $0.6t²$. By setting these two expressions equal ($12t = 0.6t²$), you are mathematically solving for the exact moment the car's accelerating pace compensates for the bus's head start. Solving for $t$ gives (C) 20 s.

UPSC often includes "distractor" options to catch common conceptual errors. For instance, (D) 12 s is a classic trap for students who might mistakenly equate the numerical value of speed with time without calculating the physics. (B) 8 s might be reached if a student incorrectly uses the formula for final velocity ($v = u + at$) instead of displacement. Always remember: to find when they meet, you must compare where they are (displacement), not how fast they are going (velocity). Mastering this distinction is key to scoring in the CSAT and General Science papers.