Two trains $ 135 \text{ m}$ and $ 155 \text{ m}$ in length, re... | Mathematics MCQ

MathematicsPrevious Year Paper 2024

Two trains $135 \text{ m}$ and $155 \text{ m}$ in length, respectively, are running in opposite directions, one at the rate of $80 \text{ km/h}$ and the other at the rate of $65 \text{ km/h}$ . In how much time will they be completely clear of each other from the moment they meet?

7.5 sec

6.8 sec

6.5 sec

7.2 sec

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

7.2 sec

Quick Summary:

Sum the lengths (290 m), sum the relative speeds (145 km/h), convert relative speed to m/s by multiplying 145 by 5/18 (~40.27 m/s), then divide 290 by the result.

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📐MAMath SolutionDirect Formula

📋 Given

Length of train 1 = 135 m, Length of train 2 = 155 m, Speed of train 1 = 80 km/h, Speed of train 2 = 65 km/h, Trains running in opposite directions.

🔢 Formula Used

t=L1+L2(S1+S2)×518t = \frac{L_1 + L_2}{(S_1 + S_2) \times \frac{5}{18}}t=(S1​+S2​)×185​L1​+L2​​

⚡ Exam Hall Shortcut / Speed Trick

Sum the lengths (290 m), sum the relative speeds (145 km/h), convert relative speed to m/s by multiplying 145 by 5/18 (~40.27 m/s), then divide 290 by the result.

⚠️ Common Student Trap / Pitfall

Failing to convert relative speed from km/h to m/s by multiplying by 5/18, or failing to add the lengths of both trains.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate Total Distance and Relative Speed

Total distance covered is the sum of both train lengths: $135 + 155 = 290 \text{ m}$ . Since the trains are moving in opposite directions, relative speed is the sum of individual speeds: $80 + 65 = 145 \text{ km/h}$ .

Dtotal=135+155=290 m,Srel=80+65=145 km/hD_{total} = 135 + 155 = 290 \text{ m}, \quad S_{rel} = 80 + 65 = 145 \text{ km/h}Dtotal​=135+155=290 m,Srel​=80+65=145 km/h

Convert Relative Speed to m/s

To convert the relative speed from km/h to m/s, multiply by $5/18$ : $145 \times \frac{5}{18} = \frac{725}{18} \text{ m/s}$ .

Srel=145×518=72518 m/sS_{rel} = 145 \times \frac{5}{18} = \frac{725}{18} \text{ m/s}Srel​=145×185​=18725​ m/s

Calculate Time

Time taken is distance divided by speed: $t = 290 / (725/18) = 290 \times (18/725)$ . Simplifying, $290 \times 18 = 5220$ ; $5220 / 725 = 7.2 \text{ seconds}$ .

t=290×18725=7.2 sect = \frac{290 \times 18}{725} = 7.2 \text{ sec}t=725290×18​=7.2 sec

✅

D is correct because the relative speed of $145 \text{ km/h}$ converts to $\approx 40.27 \text{ m/s}$ , and dividing the total length of $290 \text{ m}$ by this speed yields exactly $7.2 \text{ seconds}$ .

Core Concepts Used

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Relative Speed Unit Conversion Time-Distance-Speed Relationship

💡 EXAM TIP

This concept of relative speed is fundamental in kinematics; remember that opposite directions imply summation of speeds ( $v_1 + v_2$ ) while same-direction motion implies subtraction ( $v_1 - v_2$ ).

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