Intermediate

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Speed, Time & Distance: Formulas, Relative Speed & Trains

Master Speed, Time & Distance for UPSC CSAT, CAT, and GMAT. Comprehensive study notes covering average speed, relative speed, train crossings, and boats & streams.

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The concepts of Speed, Time & Distance form the core of the Quantitative Aptitude section in exams like CAT, GMAT, UPSC CSAT, and banking exams. Mastering these concepts requires a strong grasp of proportionalities, relative speeds, and unit conversions.

Speed is defined as the distance covered by an object in unit time. The fundamental equation is:

S p e e d = \frac{D i s t a n c e}{T i m e}

1. Basic Conversions

km/h to m/s and vice-versa

Unit conversion is the most common trap in aptitude questions. Always ensure your units (meters, kilometers, seconds, hours) match before applying formulas.

To convert speed from km/h to m/s, multiply by $\frac{5}{18}$ .
To convert speed from m/s to km/h, multiply by $\frac{18}{5}$ .

Memory Tip: When converting to a smaller unit (m/s), the smaller number (5) is on top. When converting to a larger unit (km/h), the larger number (18) is on top.

2. Average Speed

Generally, a vehicle does not cover the entire distance at a uniform speed. In such cases, average speed is not the simple average of all speeds.

Average Speed = \frac{Total Distance Covered}{Total Time Taken}

Shortcut for Equal Distances:

If a body travels equal distances with speeds $x$ and $y$ , the average speed is the harmonic mean of the speeds:

Average Speed = \frac{2 x y}{x + y}

3. Relative Speed

Objects in Motion

The relative speed of a moving body with speed $x$ km/h in relation to another body moving with speed $y$ km/h is calculated as follows:

Same Direction: Difference of their speeds i.e., $(x - y)$ km/h, where $x > y$ .
Opposite Direction: Sum of their speeds i.e., $(x + y)$ km/h.

The "Early/Late" Distance Formula

If a body travels at a speed of $x$ km/h and reaches the destination late by $t_{1}$ minutes, but traveling at $y$ km/h it reaches $t_{2}$ minutes earlier, the total distance $D$ is:

D = \frac{x \cdot y}{y - x} \times (\frac{t_{1} + t_{2}}{60})

Example: If Aman cycles at 10 km/h, he reaches late by 4 min. At 12 km/h, he reaches early by 2 min. Distance = $\frac{10 \times 12}{12 - 10} \times \frac{4 + 2}{60} = \frac{120}{2} \times \frac{6}{60} = 6 km$ .

4. Motion of Trains

Train Crossing Scenarios

Crossing a Stationary Object (No Length): Time taken by a train $x$ meters long in passing a pole or standing man = Time taken to cover $x$ meters.
Crossing a Stationary Object (With Length): Time taken by a train $x$ meters long in passing a bridge/tunnel of length $y$ meters = Time taken to cover $(x + y)$ meters.
Two Trains Crossing (Opposite): Trains of length $x$ and $y$ moving at speeds $u$ and $v$ . Time = $\frac{x + y}{u + v}$ .
Two Trains Crossing (Same): Trains of length $x$ and $y$ moving at speeds $u$ and $v$ ( $u > v$ ). Time = $\frac{x + y}{u - v}$ .

Post-Crossing Speeds Shortcut:

If two trains $A$ and $B$ start at the same time from points $P$ and $Q$ towards each other, and after crossing they take $a$ and $b$ hours to reach their destinations. Then:

\frac{Speed of A}{Speed of B} = \sqrt{\frac{b}{a}}

5. Boats and Streams

The direction along the stream is called downstream, and against the stream is upstream.

Let speed of boat in still water = $x$ km/h. Let speed of stream = $y$ km/h.
Downstream Speed ( $u$ ): $(x + y)$ km/h.
Upstream Speed ( $v$ ): $(x - y)$ km/h.

Finding Base Speeds from Down/Up Speeds:

If downstream speed is $u$ and upstream speed is $v$ :

Speed of boat in still water = $\frac{u + v}{2}$ .
Speed of stream = $\frac{u - v}{2}$ .

Solved Examples

Q1: Train in a Tunnel
A train travelling at 90 km/h passes through a tunnel 675 m long. Find how long a passenger travelling by the train remains inside the tunnel.

Solution:
Speed = $90 km/h = 90 \times \frac{5}{18} = 25 m/s$ .
The passenger remains inside until the train covers the distance equal to the length of the tunnel (675 m). Note: We do not add the train's length because we are tracking a single passenger (a point object), not the whole train.
Time = $\frac{675}{25} = 27 seconds$ .

Q2: Man Rowing
A man can row 20 km/h in still water. If it takes him thrice as long to row up as to row down the river. Find the speed of the stream.

Solution:
Ratio of time taken (Up : Down) = $3 : 1$ .
Ratio of speed (Up : Down) = $1 : 3$ .
Let upstream speed $v = x$ , then downstream speed $u = 3 x$ .
Speed in still water = $\frac{u + v}{2} = \frac{3 x + x}{2} = 2 x$ .
Given $2 x = 20 \Rightarrow x = 10 km/h$ .
Downstream speed $u = 3 x = 30 km/h$ .
Speed of stream = $\frac{u - v}{2} = \frac{30 - 10}{2} = 10 km/h$ .

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DifficultyIntermediate

Est. Read Time15 minutes

CategoryPractical Arithmetic

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