Relative velocity — two trains approaching each other on parallel tracks

Question

Two trains are moving towards each other on parallel tracks. Train A has a speed of 72 km/h and Train B has a speed of 54 km/h. Find the relative velocity of Train A with respect to Train B. If the trains are initially 500 m apart, how long will it take for them to cross each other? (Assume both trains are very short compared to the distance.)

(NCERT Class 11, Chapter 3 — Motion in a Straight Line)

Solution — Step by Step

Convert speeds to m/s

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

Train A: $72 \times \frac{5}{18} = 20$ m/s

Train B: $54 \times \frac{5}{18} = 15$ m/s

Find relative velocity

When two objects move towards each other (opposite directions), their relative velocity is the sum of their speeds.

Taking Train A's direction as positive, Train B moves in the negative direction:

$v_{A \text{ w.r.t. } B} = v_A - v_B = 20 - (-15) = 20 + 15 = \mathbf{35 \text{ m/s}}$

The trains approach each other at 35 m/s.

Time to meet

$t = \frac{\text{distance}}{\text{relative velocity}} = \frac{500}{35} = \frac{100}{7} \approx \mathbf{14.3 \text{ s}}$

Why This Works

Relative velocity asks: "How fast is A moving, as seen from B's frame of reference?" If you were sitting in Train B, you'd see Train A approaching at 35 m/s — much faster than either train's actual speed. That's because both trains are eating into the gap between them simultaneously.

The formula $v_{AB} = v_A - v_B$ works in all cases:

Objects moving towards each other: velocities have opposite signs → relative velocity = sum of speeds
Objects moving in the same direction: velocities have the same sign → relative velocity = difference of speeds
Object moving away: relative velocity is positive, meaning they're separating

Alternative Method — Frame of reference approach

Imagine you're standing still and watching both trains. In 1 second, Train A covers 20 m towards you, and Train B covers 15 m towards you. Together, they close the gap by 35 m every second. So time $= 500/35$ seconds.

💡 Expert Tip

The km/h to m/s conversion ( $\times 5/18$ ) appears in almost every kinematics problem. Memorise it cold. Quick mental math: 36 km/h = 10 m/s, 72 km/h = 20 m/s, 108 km/h = 30 m/s. For the reverse (m/s to km/h), multiply by $18/5$ or equivalently $3.6$ .

Common Mistake

⚠️ Common Mistake

Students often subtract speeds even when trains move towards each other: $20 - 15 = 5$ m/s. This gives the relative velocity for trains moving in the same direction (like one overtaking the other). For opposite directions, you must add the speeds. The sign convention handles this automatically — if you consistently define one direction as positive and assign negative velocity to the opposite direction, $v_A - v_B$ always gives the correct answer.

Question

Solution — Step by Step

Why This Works

Alternative Method — Frame of reference approach

Common Mistake

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