Relative velocity

What is relative velocity?

Finally, to understand the relative nature of motion, we introduce the concept of relative velocity.

NCERT Physics Part I (Class 11), Section 2.1, page 13

Both the JEE Main 2027 and the NEET (UG) 2026 syllabuses list relative velocity in Unit 2, Kinematics. The 2026–27 NCERT book still promises it in its introduction (above) and lists “2.5 Relative velocity” on the chapter’s first page, but the section itself is no longer in the book. What remains is NCERT Exercise 2.14 (page 25), the police-van problem solved below as Example 2, and Points to Ponder 5 in Chapter 3 (page 46). That point writes v₁₂ = v₁ − v₂ but calls it the “velocity of object 1 relative to velocity of object 2”; it should say relative to object 2. This page covers the topic in full.

In plain words: sitting in a train that is passing a slower train, you see the other train drift slowly backwards, though both are moving forwards over the ground. What you see is the relative velocity: the difference of the two velocities.

Relative velocity in one dimension

Take two objects A and B moving along the x-axis with velocities v_A and v_B (signed, measured from the ground).

• The position of B relative to A is x_B − x_A. It changes at the rate v_BA = v_B − v_A.
• Same direction: the relative speed is the absolute difference of the speeds. Two cars at 20 m/s and 15 m/s in the same direction separate (or close) at 5 m/s.
• Opposite directions: the relative speed is the sum of the speeds. The same cars approaching each other close at 35 m/s.
• If v_A = v_B, the relative velocity is zero: the two stay the same distance apart, and each sees the other at rest.

On position-time graphs, the relative velocity is the difference of the slopes of the two lines. Parallel lines mean equal velocities (zero relative velocity, constant separation); lines that cross mean the objects meet there.

Relative motion is also why the same event looks different from different frames: a ball sliding on the smooth floor of a braking train moves forward relative to the train but keeps its speed relative to the ground (see the law of inertia examples).

Relative velocity in two dimensions

When the velocities are not along one line, the subtraction is a vector subtraction: v⃗_AB = v⃗_A + (−v⃗_B). Reverse v⃗_B and add it to v⃗_A by the triangle or parallelogram rule. If the angle between v⃗_A and v⃗_B is θ:

The minus sign appears because −v⃗_B makes an angle 180° − θ with v⃗_A. For perpendicular velocities (θ = 90°) it reduces to Pythagoras’ theorem.

Rain and a moving person. Rain falling vertically at v_r looks slanted to someone moving horizontally at v_m, because what they see is v⃗_rm = v⃗_r − v⃗_m: the rain’s velocity plus a horizontal part opposite to their own motion. The rain seems to come from ahead, at an angle α to the vertical with tan α = v_m/v_r, so an umbrella should be tilted forwards by α.

NCERT’s Example 3.1 (page 31) uses the same numbers for a different situation: a boy standing still while a wind blows the rain towards the west at 12 m/s. There the rain’s ground velocity itself is slanted, so the umbrella tilts towards the east, the side the wind comes from. Rain problems always come down to the rain’s velocity relative to the person.

River crossing

A boat moves at speed u relative to the water, and the river flows at v relative to the banks. The boat’s velocity relative to the bank is the vector sum v⃗_{boat, bank} = v⃗_{boat, water} + v⃗_{water, bank}. The formulas below assume straight parallel banks a distance d apart, a steady uniform current along the banks, and a fixed boat speed u relative to the water. The boat maintains its chosen heading throughout the crossing:

• Shortest time: point the boat straight across. Only the across component carries it over, so t = d/u. Meanwhile the current carries it downstream by the drift vd/u.
• Shortest path (landing directly opposite): point the boat upstream at an angle θ to the straight-across direction, with sin θ = v/u, so that its upstream component cancels the current. Its speed across is √(u^² − v^²), and t = d/√(u^² − v^²). A finite-time crossing directly opposite is possible only if u > v. If u = v, exactly cancelling the current uses all the boat’s speed upstream and leaves zero speed across the river.
• Least drift when u < v: the boat cannot land directly opposite, but it drifts least if it heads upstream at an angle θ to the straight-across direction, with sin θ = u/v. Its velocity relative to the bank is then at right angles to its heading, and the drift is d√(v^² − u^²)/u.

Solved examples

Common mistakes to avoid

Quick revision summary

• Velocity of A relative to B: v⃗_AB = v⃗_A − v⃗_B, both measured in the same frame; v⃗_BA = −v⃗_AB.
• One dimension: same direction, speeds subtract; opposite directions, speeds add.
• Two dimensions: subtract as vectors; |v⃗_AB|^² = v_A² + v_B² − 2v_Av_B cos θ.
• Rain: use the rain’s velocity relative to the person; tan α = v_m/v_r for vertical rain.
• River: shortest time, head across, t = d/u, drift vd/u; shortest path, sin θ = v/u, t = d/√(u^² − v^²).

Questions students ask

Is relative velocity still in the JEE Main and NEET syllabus, though NCERT dropped it?

Yes, relative velocity is still in the syllabus: both the JEE Main 2027 and the NEET (UG) 2026 syllabuses name it in Unit 2: Kinematics. The 2026–27 NCERT book still lists “2.5 Relative velocity” on the first page of Chapter 2, Motion in a Straight Line, but no longer has the section; only Exercise 2.14 (page 25) and Points to Ponder 5 of Chapter 3 (page 46) still use the idea.

Does relative velocity have a unit, or is it dimensionless?

Relative velocity has the same unit and dimensions as any velocity, m s^⁻¹ in SI and [M^⁰ L T^⁻¹]: it is the difference of two velocities, v_A − v_B, not a ratio, so it is not dimensionless. The word “relative” only says from whose point of view it is measured, here an observer moving with B. Relative quantities that are ratios, such as relative density, are the ones without a unit.

What is the difference between relative speed and relative velocity?

Relative speed is the magnitude of the relative velocity, so it has no direction: the velocity of A relative to B and that of B relative to A are opposite, v_BA = −v_AB, but have the same relative speed. For two bodies with speeds v_A and v_B, the relative speed lies between |v_A − v_B|, when they move the same way, and v_A + v_B, when they move in opposite directions, so it can exceed both speeds. For two cars at 12 m s^⁻¹ and 9 m s^⁻¹, it is 3 m s^⁻¹ if they move the same way, 21 m s^⁻¹ head-on and 15 m s^⁻¹ at right angles.

What is relative acceleration, and what is it for two freely falling bodies?

The acceleration of A relative to B is a_AB = a_A − a_B, defined like relative velocity, and for two bodies freely moving in the same uniform gravitational field it is zero, because they share the same acceleration vector g (air resistance neglected). So each sees the other move in a straight line at constant velocity, which makes meeting problems quick: if a ball is dropped from a height of 40 m at the moment another is thrown straight up at 20 m s^⁻¹ from the ground directly below, they close at a steady 20 m s^⁻¹ and meet after 40/20 = 2 s. The value of g does not enter the meeting time, as long as both are still in the air.

Does relative velocity depend on the frame of reference?

No, the velocity of one body relative to another is the same in every frame of reference that moves without rotating, even an accelerating one, because changing to such a frame shifts both velocities at each instant by the frame’s own velocity, which cancels in v_A − v_B. The separate velocities do depend on the frame: a train at 25 m s^⁻¹ and a car at 18 m s^⁻¹, both heading east, move at 20 m s^⁻¹ and 13 m s^⁻¹ as seen by a cyclist riding east at 5 m s^⁻¹, yet the train gains on the car at 7 m s^⁻¹ in both frames. This simple rule holds at everyday speeds; near the speed of light, special relativity replaces it.

What are upstream and downstream speeds in river boat problems?

In river boat problems, downstream means moving with the current and upstream against it: a boat with speed u relative to the water, in a river flowing at v, moves at u + v relative to the bank downstream and at u − v upstream, which is possible only if u > v. A round trip of length L each way therefore takes L/(u + v) + L/(u − v) = 2Lu/(u^² − v^²), longer than the 2L/u it would take in still water. For example, with u = 4 m s^⁻¹, v = 2 m s^⁻¹ and L = 1200 m, the boat takes 200 s down and 600 s back, 800 s in all instead of 600 s.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Two cars move in the same direction along a straight road at 72 km/h and 54 km/h. Find the velocity of the faster car relative to the slower one.

   Show answerHide answer

   20 m/s − 15 m/s = 5 m/s (18 km/h) in the direction of motion.

2. Q2. Two toy cars move towards each other along a straight track at 8 m/s and 6 m/s, starting 28 m apart. When do they meet?

   Show answerHide answer

   Relative speed 8 + 6 = 14 m/s, so they meet after 28/14 = 2 s.

3. Q3. Why can a boat whose speed relative to the water is less than the river’s speed never land directly opposite its starting point?

   Show answerHide answer

   To land opposite, the boat’s upstream component must cancel the current, which needs u sin θ = v. If u < v, no angle can do this, so the boat always drifts downstream.