3.6 Speed, Distance, Time, Trains & Boats/Streams

Speed, Distance, Time, Trains & Boats/Streams

Kinematics word problems are central to SBI PO Quantitative Aptitude. Whether analyzing two trains crossing on parallel tracks or calculating river current velocity against a boat's rowing speed, precision in unit conversions and relative motion principles is required.

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1. Kinematics Fundamentals & Unit Conversions

Core Equation

$$\text{Distance} = \text{Speed} \times \text{Time} \implies D = S \times T$$

Unit Conversion Formulas

• To convert $\text{km/h}$ to $\text{m/s}$, multiply by $\frac{5}{18}$: $$1 \text{ km/h} = \frac{1000 \text{ meters}}{3600 \text{ seconds}} = \frac{5}{18} \text{ m/s}$$
• To convert $\text{m/s}$ to $\text{km/h}$, multiply by $\frac{18}{5}$.

Average Speed Formulas

• Equal Distances: If a body covers distance $d$ at speed $s_1$ and another equal distance $d$ at speed $s_2$, average speed is the Harmonic Mean: $$\text{Average Speed} = \frac{2 s_1 s_2}{s_1 + s_2}$$
• Equal Time Intervals: If a body travels for time $t$ at speed $s_1$ and for time $t$ at speed $s_2$, average speed is the Arithmetic Mean: $$\text{Average Speed} = \frac{s_1 + s_2}{2}$$

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2. Relative Speed & Problems on Trains

Relative Speed Principles

• Opposite Directions: When two bodies move toward each other at speeds $s_1$ and $s_2$: $$S_{\text{rel}} = s_1 + s_2$$
• Same Direction: When two bodies move in the same direction at speeds $s_1 > s_2$: $$S_{\text{rel}} = s_1 - s_2$$

Train Distance Scenarios

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3. Boats and Streams Vector Mechanics

Let speed of boat in still water $= u \text{ km/h}$, and speed of stream current $= v \text{ km/h}$.

• Downstream Speed ($D$): Moving with the stream current: $$D = u + v$$
• Upstream Speed ($U$): Moving against the stream current: $$U = u - v$$
• Speed of Boat in Still Water ($u$): $$u = \frac{D + U}{2}$$
• Speed of Stream Current ($v$): $$v = \frac{D - U}{2}$$

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4. Step-by-Step Worked Math Examples

Worked Example 1: Boat Downstream & Upstream Round Trip

Problem: A boat travels 24 km downstream in 2 hours and covers the same distance upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream current.

Step-by-Step Solution:

1. Calculate Downstream Speed $D = \frac{\text{Distance}}{\text{Time}} = \frac{24}{2} = 12 \text{ km/h}$.
2. Calculate Upstream Speed $U = \frac{\text{Distance}}{\text{Time}} = \frac{24}{4} = 6 \text{ km/h}$.
3. Speed of boat in still water $u = \frac{D + U}{2} = \frac{12 + 6}{2} = \frac{18}{2} = 9 \text{ km/h}$.
4. Speed of stream current $v = \frac{D - U}{2} = \frac{12 - 6}{2} = \frac{6}{2} = 3 \text{ km/h}$.

Worked Example 2: Two Trains Crossing Each Other

Problem: Two trains of lengths 140 meters and 160 meters are running on parallel tracks in opposite directions at speeds of 60 km/h and 48 km/h respectively. In how many seconds will they cross each other completely?

Step-by-Step Solution:

1. Total distance to be covered $D = L_1 + L_2 = 140 + 160 = 300 \text{ meters}$.
2. Since moving in opposite directions, relative speed $S_{\text{rel}} = S_1 + S_2 = 60 + 48 = 108 \text{ km/h}$.
3. Convert relative speed to m/s: $108 \times \frac{5}{18} = 6 \times 5 = 30 \text{ m/s}$.
4. Time required to cross $= \frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{300}{30} = 10 \text{ seconds}$.

Worked Example 3: Train Crossing Platform & Moving Man

Problem: A train running at 72 km/h crosses a 200-meter long platform in 22 seconds. How long will it take to cross a man running at 18 km/h in the same direction?

Step-by-Step Solution:

1. Convert train speed: $72 \times \frac{5}{18} = 20 \text{ m/s}$.
2. Let length of train be $L$.
3. When crossing platform, distance $= L + 200$, time $= 22 \text{ s}$.
4. $L + 200 = \text{Speed} \times \text{Time} = 20 \times 22 = 440 \text{ m} \implies L = 440 - 200 = 240 \text{ meters}$.
5. Convert man's speed: $18 \times \frac{5}{18} = 5 \text{ m/s}$.
6. Relative speed when moving in same direction $= 20 - 5 = 15 \text{ m/s}$.
7. Distance to cross man $= L = 240 \text{ meters}$.
8. Time taken to cross man $= \frac{240}{15} = 16 \text{ seconds}$.

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5. Exam Pitfalls & Calculation Traps

• Forgetting Unit Conversion: Never mix km/h with meters or seconds. Convert all quantities to m/s and meters before calculating time.
• Ignoring Platform Length: When a train crosses a bridge or tunnel, distance is $L_{\text{train}} + L_{\text{platform}}$, not just $L_{\text{platform}}$.