Relative Speed Concepts and Formulas for CMAT

2026-06-27

Introduction to Relative Speed

Relative speed is the speed of one moving body with respect to another moving body. In the CMAT Quantitative Ability section, questions based on Time, Speed, and Distance frequently involve multiple moving objects. Understanding how to calculate their combined speed relative to one another helps you solve complex scenarios—such as trains overtaking each other, police-thief chases, or objects moving in opposite directions—in a matter of seconds.

Core Principles of Relative Speed

When calculating relative speed, the direction of motion of both objects is the single most critical factor. There are two primary scenarios:

Case 1: Objects Moving in the Same Direction

When two objects are moving in the same direction, the effective speed is the difference between their individual speeds. The faster object is catching up to or moving away from the slower object.

• Let the speed of the faster object be $S_1$ and the slower object be $S_2$.
• Relative Speed ($S_R$) $= S_1 - S_2$

Case 2: Objects Moving in Opposite Directions

When two objects are moving in opposite directions (either towards each other or away from each other), they cover the relative distance much faster because their motions complement each other. The effective speed is the sum of their individual speeds.

• Let the speed of the first object be $S_1$ and the second object be $S_2$.
• Relative Speed ($S_R$) $= S_1 + S_2$

Summary of Formulae

Important Unit Conversions

CMAT questions often mix units (e.g., speed in km/hr and distance in meters). You must convert them to a uniform system before applying relative speed formulas:

• To convert km/hr to m/s: Multiply by $\frac{5}{18}$ $$S\text{ (m/s)} = S\text{ (km/hr)} \times \frac{5}{18}$$
• To convert m/s to km/hr: Multiply by $\frac{18}{5}$ $$S\text{ (km/hr)} = S\text{ (m/s)} \times \frac{18}{5}$$

Solved Examples

Example 1 (Same Direction)

Question: A thief steals a car at $2:00\text{ PM}$ and drives it at $60\text{ km/hr}$. The theft is discovered at $2:30\text{ PM}$ and the owner sets off in another car at $75\text{ km/hr}$ to catch him. At what time will the owner catch the thief?

Step-by-step Solution:

1. Find the head start distance: The thief drove alone for $30\text{ minutes}$ ($0.5\text{ hours}$) before the owner started chasing. $$\text{Distance covered by thief} = \text{Speed} \times \text{Time} = 60 \times 0.5 = 30\text{ km}$$
2. Calculate Relative Speed: Since both are moving in the same direction: $$S_R = 75 - 60 = 15\text{ km/hr}$$
3. Calculate Time to Catch: The owner needs to cover the $30\text{ km}$ gap with the relative speed of $15\text{ km/hr}$. $$\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{30\text{ km}}{15\text{ km/hr}} = 2\text{ hours}$$
4. Find the Meeting Time: The chase started at $2:30\text{ PM}$. $$\text{Meeting Time} = 2:30\text{ PM} + 2\text{ hours} = 4:30\text{ PM}$$

Example 2 (Opposite Direction)

Question: Two stations $A$ and $B$ are $450\text{ km}$ apart. A train starts from $A$ at $8:00\text{ AM}$ towards $B$ at $50\text{ km/hr}$. Another train starts from $B$ at $9:00\text{ AM}$ towards $A$ at $30\text{ km/hr}$. At what time will they meet?

Step-by-step Solution:

1. Equalize the starting time: The first train starts at $8:00\text{ AM}$ and the second starts at $9:00\text{ AM}$. We must find how much distance the first train covers in that $1\text{ hour}$ gap. $$\text{Distance covered by Train 1 in 1 hour} = 50\text{ km/hr} \times 1\text{ hr} = 50\text{ km}$$
2. Find the remaining distance: $$\text{Remaining Distance} = 450 - 50 = 400\text{ km}$$
3. Calculate Relative Speed: Since they are moving towards each other (opposite directions): $$S_R = 50 + 30 = 80\text{ km/hr}$$
4. Calculate Time to Meet: $$\text{Time} = \frac{\text{Remaining Distance}}{S_R} = \frac{400}{80} = 5\text{ hours}$$
5. Find the Meeting Time: Count $5\text{ hours}$ from the common starting time ($9:00\text{ AM}$). $$\text{Meeting Time} = 9:00\text{ AM} + 5\text{ hours} = 2:00\text{ PM}$$

Common Traps to Avoid

• Starting Time Mismatch: Never apply relative speed formulas directly if the two objects start moving at different times. Always calculate the distance covered by the first moving object first, and then apply the formula to the remaining distance from the moment both are in motion.
• Incorrect Unit Matching: Ensure that all values (speed, distance, and time) are in compatible units. Convert $\text{km/hr}$ to $\text{m/s}$ or vice versa before starting your calculations.
• Relative Speed Sign Confusion: Remember: Same direction is subtraction ($-$); opposite direction is addition ($+$). A common error is swapping these signs under pressure.

Practise these as timed MCQs to lock the pattern in.