Two cars X and Y start from two places A and B respectively which are 700 km apart at 9 a.m. Both the cars run at an average speed of 60 km/hr. Car X stops at 10 a.m. and again starts at 11 a.m. while the other-car Y continues to run without stopping. When do the two cars cross each other?

1. The Core Mechanics: Speed, Distance, and Time (basic)

Welcome! We are starting at the very root of Quantitative Aptitude: the relationship between Speed, Distance, and Time. At its simplest, Speed is the rate at which an object covers distance. As defined in Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113, speed is the distance covered by an object in a unit time, such as one second, one minute, or one hour. It tells us how fast one object is moving compared to another.

To master this, you must become comfortable with the fundamental formula. If you know any two variables, you can always find the third. These relationships are expressed as:

• Speed = Total Distance ÷ Total Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

As noted in Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115, if a bus moves at a constant speed of 50 km/h for 2 hours, it will cover exactly 100 km. However, in real-world scenarios, objects rarely move at a perfectly constant speed. This leads us to the concept of Average Speed, which is the total distance divided by the total time taken, regardless of stops or changes in pace along the way.

We also distinguish between two types of motion based on consistency:

Type of Motion	Description	Example
Uniform Motion	An object covers equal distances in equal intervals of time.	A train moving at a constant 60 km/h on a straight track (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117).
Non-Uniform Motion	The speed of an object changes as it moves.	A car moving through city traffic, stopping and starting.

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

2. Unit Conversions and Time Arithmetic (basic)

To conquer Quantitative Aptitude, we must first become fluent in the language of measurement. Time, unlike our standard decimal system, operates on a sexagesimal (base-60) system. As established in Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111, the standard unit of time is the second (s). Larger units include the minute (min) and hour (h), where 60 seconds = 1 minute and 60 minutes = 1 hour. When performing time arithmetic, always remember that "1.5 hours" does not mean 1 hour and 50 minutes; it means 1 hour and 30 minutes (0.5 of 60).

Speed is the bridge between distance and time. While the SI unit is metres per second (m/s), we frequently use kilometres per hour (km/h) for transport and infrastructure contexts, such as the Indian Railways' push for 160–200 km/h on trunk routes Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.413. For maritime and aviation navigation, we even use knots, where 1 knot equals approximately 1.852 km/h Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Tropical Cyclones, p.372.

Finally, understanding the relationship between motion and time is even relevant to our planet's rotation. Since the Earth completes a 360° rotation in 24 hours, it passes through 15° of longitude every hour, or 1° every 4 minutes Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.11. Whether you are calculating the meeting time of two trains or the local time in Greenwich, the ability to convert these units accurately is your foundational tool.

Unit A	Conversion Factor	Unit B
1 Hour	× 60	Minutes
1 km/h	× 5/18	m/s
1 Knot	× 1.852	km/h

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111, 113; Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.413; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Tropical Cyclones, p.372; Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.11

3. The Concept of Relative Speed (intermediate)

Welcome back! Today we are diving into a concept that simplifies complex motion problems into one elegant calculation: Relative Speed. Simply put, relative speed is the speed of one object as observed from another moving object. Think of it as the rate at which the gap between two objects is either closing or widening. In our daily lives, we experience this constantly. For instance, when you are on a merry-go-round turning anti-clockwise, the stationary trees outside appear to be moving clockwise around you Science-Class VII, Earth, Moon, and the Sun, p.170. This shift in perspective is the essence of relative motion.

To master this for the UPSC CSAT or any aptitude test, you only need to understand two primary scenarios based on the direction of travel. When two objects move toward each other (opposite directions), they are effectively "working together" to cover the distance between them, so their speeds are added. Conversely, when they move in the same direction—like a police car chasing a thief—the relative speed is the difference between their individual speeds because the lead object is trying to "escape" the gap being closed by the follower.

Scenario	Direction	Relative Speed Formula	Conceptual Logic
Case 1	Opposite Directions (Towards or Away)	V₁ + V₂	The distance between them changes rapidly as both contribute to the movement.
Case 2	Same Direction (Chasing or Following)	V₁ – V₂ (Faster – Slower)	The distance changes slowly because one object is "running away" from the other.

Once you have determined the relative speed, you can treat the problem as a simple distance-time calculation where Time = Distance / Relative Speed. Note that in most competitive exam problems, we assume a constant or average speed throughout the journey Science-Class VII, Measurement of Time and Motion, p.115. If one object stops for a while, the relative speed logic only applies during the intervals when both objects are in motion. During the period one is stationary, the gap closes only at the speed of the active object.

Sources: Science-Class VII, Earth, Moon, and the Sun, p.170; Science-Class VII, Measurement of Time and Motion, p.115

4. Connected Concept: Problems on Trains (intermediate)

To master problems on trains, we must first ground ourselves in the fundamental motion equation: Speed = Distance / Time. In most competitive exams, the primary challenge arises from unit consistency. While speeds are often given in km/h, distances like train lengths or platform dimensions are given in metres. To convert km/h to m/s, we multiply by 5/18; to convert m/s to km/h, we multiply by 18/5 Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118. For instance, a train on a Broad Gauge track (which has a width of 1.676 metres) is treated as having a specific length that must be added to the distance whenever it crosses a stationary object like a bridge or a platform INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Transport and Communication, p.79.

The concept of Relative Speed is the heart of train problems. When two objects move toward each other (opposite directions), their speeds are additive because the distance between them closes at a faster rate. Conversely, if they move in the same direction, the relative speed is the difference between the two. However, real-world scenarios often involve disruptions—such as a train stopping at a station or a car breaking down. In these cases, we calculate the distance covered during the period of uniform motion first, then reassess the remaining distance and the new relative speed once the stopped object resumes its journey.

When a train crosses a stationary point (like a pole or a signal), the distance covered is simply the length of the train. However, when it crosses a stationary object with length (like a platform or another stationary train), the total distance is the sum of both lengths. This is because the 'crossing' is only complete when the tail of the train clears the end of the object Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115.

Scenario	Relative Speed Formula	Distance to be Covered
Opposite Directions	S₁ + S₂	L₁ + L₂
Same Direction	S₁ - S₂	L₁ + L₂
Crossing a Pole	S₁	L₁

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115, 118; INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Transport and Communication, p.79

5. Connected Concept: Boats and Streams (intermediate)

In quantitative aptitude, the concept of Boats and Streams is a beautiful application of relative motion. Unlike a vehicle moving on a static road, a boat moves on a medium (water) that is itself in motion. To understand this, we first look at the drift or the speed of the current. In physical geography, we define the 'strength' of a current simply as its speed FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.111. When you are rowing, your actual progress depends on whether that drift is helping you or hindering you.

We classify the direction of movement into two categories based on the flow of the water:

• Downstream: This is when the boat moves with the flow of the stream. Here, the water acts as a booster. If your boat's speed in still water is 'u' and the stream's speed is 'v', your effective speed becomes u + v.
• Upstream: This is when the boat moves against the flow. The water acts as resistance. Just as in supply chains where 'upstream' refers to moving back toward the source Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.363, in a river, it means fighting the current. Your effective speed is u - v.

To master these problems, you must be comfortable switching between the speed of the boat in still water and the speeds relative to the stream. The fundamental formula of motion—where speed is the distance covered in unit time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115—remains your primary tool. However, you must always adjust the 'speed' variable based on the direction of the flow.

Scenario	Formula	Logic
Downstream Speed (D)	u + v	Boat and stream move in the same direction.
Upstream Speed (U)	u - v	Boat moves against the direction of the stream.
Speed in Still Water (u)	(D + U) / 2	The average of the two effective speeds.
Speed of Stream (v)	(D - U) / 2	Half the difference between downstream and upstream speeds.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.111; Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.363; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115

6. Managing Stoppages and Time Offsets (exam-level)

To master complex motion problems, we must first understand Relative Speed. When two objects move toward each other, their speeds are additive (v₁ + v₂). However, real-world scenarios—much like the staggered time zones across the Trans-Siberian Railway Certificate Physical and Human Geography, The Earth's Crust, p.13—often involve disruptions where one object stops while the other continues. To solve these, we don't use a single formula; instead, we use a 'Time-Slice' approach. We break the journey into intervals based on when the state of motion changes, ensuring we account for every minute of elapsed time.

The core challenge is managing the Time Offset—the period when only one object is moving. During a stoppage, the relative speed effectively drops from (v₁ + v₂) to just the speed of the active object. This is conceptually similar to how Daylight Saving Time shifts the start of a workday Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.254; the total 'distance' of the day remains the same, but the 'active' hours shift. In aptitude, we calculate the distance covered during this offset and subtract it from the total distance before reapplying the combined relative speed for the final approach.

By using the fundamental relationship that Time = Distance / Speed Science-Class VII, Measurement of Time and Motion, p.115, we can navigate any stoppage. You simply find the distance remaining at the exact moment the stopped object resumes its journey. From that timestamp onwards, both objects are closing the gap together again.

Scenario Phase	Speed Logic	Calculation Step
Both Moving	Relative Speed (v₁ + v₂)	Subtract (v₁ + v₂) × time from total distance.
One Stays/Stops	Single Speed (v₁ or v₂)	Subtract moving object's distance from remaining gap.
Both Resume	Relative Speed (v₁ + v₂)	Divide final remaining distance by (v₁ + v₂).

Sources: Certificate Physical and Human Geography, The Earth's Crust, p.13; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.254; Science-Class VII, Measurement of Time and Motion, p.115

7. Solving the Original PYQ (exam-level)

This problem is a masterclass in applying Relative Speed within the framework of Segmented Motion. You have already learned that when two objects move toward each other, their speeds are additive ($S_1 + S_2$). However, the UPSC often introduces a "pause" or a "delayed start" to test your ability to break a complex journey into manageable time intervals. To solve this, you must calculate the remaining distance at each transition point—specifically at 10 a.m. when Car X stops and at 11 a.m. when it resumes—before applying the relative speed formula to the final stretch.

Let’s walk through the logic: From 9 a.m. to 10 a.m., both cars move, covering a combined $60 + 60 = 120$ km, which leaves 580 km between them. Between 10 a.m. and 11 a.m., only Car Y is moving, covering an additional 60 km alone. This leaves a gap of 520 km at the 11 a.m. mark. From this point forward, both cars are moving again with a combined relative speed of 120 km/h. Dividing the remaining 520 km by 120 km/h gives us $4 rac{1}{3}$ hours, or 4 hours and 20 minutes. Adding this duration to the 11 a.m. restart time leads us directly to the correct answer: 3:20 p.m.

The incorrect options are classic UPSC traps designed to catch students who take shortcuts. Option (D) 4:20 p.m. is a common distractor for students who calculate the 4 hours and 20 minutes correctly but accidentally add it to the 12:00 mark or miscount the initial two hours. Option (A) 2:40 p.m. often results if a student forgets to account for the one-hour stop of Car X entirely. Always remember: in CSAT, precision in the timeline is just as important as the mathematical formula itself.