Relative Velocity

State a velocity and you have quietly chosen an observer: "the train moves at 60 km h⁻¹" means 60 km h⁻¹ as seen from the platform. Seen from another train running alongside, the same train may look frozen — or, if it approaches head-on, like a 100 km h⁻¹ bullet. Relative velocity is the bookkeeping rule that converts between observers, and it is the closing idea of this chapter because it needs everything built so far: signed 1D velocities, displacement bookkeeping, and constant-rate meeting problems. NTA draws on it for approach-speed integer questions, train and car chases, and the occasional Assertion–Reason on the definition itself. The chapter ends here, but the idea does not: the same subtraction rule, upgraded to vectors, powers projectile frames and river-boat problems in Motion in a Plane — which is why the last section of this page plants that bridge deliberately.

What is Relative Velocity? — Complete Theory

1. Definition and its one-line derivation

Let two objects A and B move along the same line with constant velocities v_A and v_B (signed, measured in the ground frame, sharing t = 0 at the same origin). Their positions are x_A = v_At and x_B = v_Bt, so their separation changes as

x_A − x_B = (v_A − v_B)t → v_AB = v_A − v_B

That is the entire theory: the velocity of A as seen from B is A's velocity minus B's velocity — with signs, in one dimension. Dimensional check: [v_A] − [v_B] = [L T⁻¹] ✔. The subscript order matters and is read "v of A with respect to B"; reversing it flips the sign, since v_BA = −v_AB. A useful identity chain for three observers: v_AC = v_AB + v_BC — relative velocities stack like vectors, a fact that becomes essential in 2D next chapter.

2. The two 1D cases: chase and head-on

One formula covers both — subtract signed velocities — but the memory-friendly summary is: same way → subtract the speeds; opposite ways → add the speeds. The quantity |v_AB| is called the speed of approach when the separation is shrinking and the speed of separation when it is growing. Meeting/catching times follow immediately: time to meet = initial gap ÷ approach speed. A head-on pair closes the gap at the sum of speeds — which is why oncoming traffic feels so much faster than overtaking traffic, an everyday observation NTA has turned into questions.

3. Worked micro-cases you should recognise on sight

Two trains crossing each other: if lengths L₁ and L₂ must pass, the crossing time is (L₁ + L₂)/|v_rel| — relative velocity supplies the rate, the combined length supplies the distance. Catching a departing bus: runner at constant speed, bus accelerating from rest — write both positions from the same origin and set them equal (a full worked version is in the chapter test, Set B). Twins on parallel tracks: same speed, same direction → v_rel = 0, each frozen for the other regardless of how fast the ground rushes by. Each case is just "choose the observer, subtract, then apply x = vt or the kinematic equations in the new frame" — the equations of motion are valid in any frame moving at constant velocity, a deep fact you will meet again as Galilean relativity.

Visualising Relative Velocity: Signed Bars

Solved Examples on Relative Velocity (Step-by-Step)

Solved Example 1 — Two trains, both cases

Q. Train A runs at 54 km h⁻¹ and train B at 36 km h⁻¹ on parallel straight tracks. Find the speed of A relative to B when (i) they run the same way and (ii) they run opposite ways.

Step 1. Convert both: 54 × 5/18 = 15 m s⁻¹; 36 × 5/18 = 10 m s⁻¹.

Step 2. Same direction: v_A = +15, v_B = +10 → v_AB = 15 − 10 = 5 m s⁻¹ (a gentle overtake).

Step 3. Opposite directions: v_A = +15, v_B = −10 → v_AB = 15 − (−10) = 25 m s⁻¹ (the angry head-on pass).

Solved Example 2 — Time to meet

Q. Two cars 500 m apart drive toward each other along a straight road at 20 m s⁻¹ and 30 m s⁻¹. After how long do they meet, and how far has the slower car moved?

Step 1. Approach speed: 20 + 30 = 50 m s⁻¹ (head-on → speeds add).

Step 2. Time to meet: t = gap/approach = 500/50 = 10 s.

Step 3. Slower car's share: 20 × 10 = 200 m (the faster covers the remaining 300 m; total 500 m ✔).

Practice Questions on Relative Velocity (With Solutions)

Practice Q1 (Numerical-value type)

Q. A cyclist at 8 m s⁻¹ chases a scooterist at 6 m s⁻¹ along the same straight road, 40 m ahead. The time to catch up is ______ s.

Solution. Chase → relative speed = 8 − 6 = 2 m s⁻¹; t = 40/2 = 20 s. (Same-way motion subtracts speeds.)

Practice Q2 (Single-correct MCQ)

Q. Two trains run at 72 km h⁻¹ and 54 km h⁻¹ toward each other on parallel tracks. Their relative speed in m s⁻¹ is: (a) 5 (b) 20 (c) 35 (d) 45

Solution. Convert: 72 → 20 m s⁻¹, 54 → 15 m s⁻¹. Head-on: 20 + 15 = 35 m s⁻¹. (Option (b) forgets the second train; (a) is the same-direction difference in km/h units gone wrong.)

Practice Q3 (Statement type)

Q. Statement I: When two bodies move in opposite directions along a line, the magnitude of their relative velocity equals the sum of their speeds. Statement II: The relative speed of two bodies is always the difference of their speed magnitudes.

Solution. I is true: opposite signs turn v_A − v_B into a sum. II is false — it fails precisely in the head-on case. So I is correct and II is incorrect.

Key Formulas & Takeaways

FAQs on Relative Velocity

What is relative velocity?

The velocity of one object as measured from another: v_AB = v_A − v_B in vector form. In one dimension the same formula works directly with signed velocities.

How do you find the relative speed of two objects moving in opposite directions?

Add the speeds: for 15 m/s and 5 m/s head-on, each appears to the other at 20 m/s. In the same direction the relative speed is the difference, 10 m/s.

When do two objects moving toward each other meet?

When the initial separation is closed at the approach speed: t = separation/(v₁ + v₂) for head-on motion, or separation/(v₁ − v₂) for a chase.

Why does rain appear slanted to a walking person?

The rain's velocity relative to the person is v_rain − v_man, which gains a backward horizontal component equal to the walking speed. The umbrella must tilt by tan⁻¹(v_man/v_rain) — the full 2D treatment comes in Motion in a Plane.