Relative Velocity

Motion is always measured relative to something. When you sit on a train and watch a platform slide past, you are observing motion from a particular reference frame — the train. A reference frame is simply a coordinate system attached to an observer, used to assign positions and times to events. The velocity an object appears to have depends entirely on which reference frame you are measuring from. This is not a trick or an illusion; it is a fundamental feature of how physicists describe the physical world, and getting it right is essential in engineering, navigation, and everyday problem-solving. Relative velocity is the velocity of one object as measured by an observer who may themselves be moving. In one dimension this amounts to simple addition or subtraction of speeds. In two dimensions, vectors must be combined using vector addition. The mathematical rules that connect measurements in one reference frame to measurements in another — at speeds much less than the speed of light — are called the Galilean transformations, named after Galileo Galilei. These transformations are the backbone of classical kinematics and remain highly accurate for all everyday speeds. Understanding relative velocity allows you to solve problems such as: how fast does a swimmer need to aim upstream to cross a river straight across? What is the velocity of an aircraft relative to the ground when wind is blowing? These questions appear in examinations and in real engineering contexts. Mastering the vector approach and the Galilean transformation equations gives you a reliable toolkit for all such scenarios.

Key takeaways
  • Reference Frames and Notation
  • Relative Velocity in One Dimension
  • Relative Velocity in Two Dimensions
  • Galilean Transformations
  • The Vector Triangle Method

Reference Frames and Notation

A reference frame S is attached to one observer, and a frame S′ is attached to a second observer moving at constant velocity V relative to S. We label the velocity of object A as seen from frame B using the notation v_{AB}, read as 'the velocity of A relative to B'. This double-subscript notation keeps bookkeeping clear: the first subscript tells you what is moving, the second tells you who is watching.

Relative Velocity in One Dimension

In 1D, if car A moves at +30 m/s (east) and car B moves at +20 m/s (east), then the velocity of A relative to B is v_{AB} = v_A − v_B = 30 − 20 = +10 m/s. Car A appears to pull away from B at 10 m/s. If B were moving west at −20 m/s, then v_{AB} = 30 − (−20) = 50 m/s. The rule is always: subtract the velocity of the observer from the velocity of the observed object. Think of it as asking 'if I zero out my own motion, how fast does the other thing move?'

Relative Velocity in Two Dimensions

In 2D each velocity is a vector with x- and y-components. The same rule applies component by component: v⃗_{AB} = v⃗_A − v⃗_B. A classic example is a river crossing: a boat points directly across a river (north) at 4 m/s relative to the water, but the current flows east at 3 m/s relative to the ground. The boat's velocity relative to the ground is the vector sum: 4 m/s north + 3 m/s east. Its speed over the ground is √(4² + 3²) = 5 m/s, and it drifts downstream at an angle arctan(3/4) ≈ 37° from north.

Galilean Transformations

Consider frame S′ moving at constant velocity V along the x-axis relative to frame S. If at t = 0 the origins coincide, the Galilean transformation equations are: x′ = x − Vt, y′ = y, z′ = z, t′ = t. Differentiating the position equation with respect to time gives the velocity transformation: u′ = u − V, where u is the object's velocity in S and u′ is its velocity in S′. These equations assume that time is absolute (the same in all frames), which is an excellent approximation for speeds much less than the speed of light (c ≈ 3 × 10⁸ m/s). At high speeds, Einstein's special relativity replaces Galilean transformations, but for A-level physics the Galilean rules are exact enough.

The Vector Triangle Method

A practical strategy for 2D relative velocity problems is to draw a vector triangle. Write down the relationship v⃗_{object/ground} = v⃗_{object/medium} + v⃗_{medium/ground}, then arrange these three vectors tip-to-tail. The unknown vector closes the triangle. You can then solve using Pythagoras (for right-angled triangles) or the sine and cosine rules (for oblique triangles). Always assign a clear positive direction convention before you start, and double-check that your subscripts chain correctly: the inner subscripts must match and cancel.

The Key Equation

v⃗AB=v⃗A−v⃗B\vec{v}_{AB} = \vec{v}_{A} - \vec{v}_{B}
  • \vec{v}_{AB} — Relative velocity of A with respect to B (m s⁻¹): The velocity that observer B measures for object A; it is a vector quantity with both magnitude and direction.
  • \vec{v}_{A} — Velocity of A in the ground frame (m s⁻¹): The velocity of object A as measured from a fixed (or chosen common) reference frame.
  • \vec{v}_{B} — Velocity of B in the ground frame (m s⁻¹): The velocity of observer B as measured from the same fixed reference frame. Subtracting this 'zeros out' B's own motion.

Worked Example

Problem. An aircraft heads due north at 250 m/s relative to the air. A wind blows from west to east at 60 m/s relative to the ground. (a) Find the velocity of the aircraft relative to the ground (magnitude and direction). (b) If a pilot wants to travel due north relative to the ground, at what angle west of north must the aircraft point, and what will its ground speed be?

∣v⃗ground∣≈257 m s−1,ϕ=arcsin⁡ ⁣(60250)≈13.9∘|\vec{v}_{\text{ground}}| \approx 257\text{ m s}^{-1},\quad \phi = \arcsin\!\left(\tfrac{60}{250}\right) \approx 13.9^\circ

Answer. Part (a): 257 m/s at 13.5° east of north. Part (b): Point 13.9° west of north; ground speed ≈ 243 m/s.

Figure
Diagram
Relative Velocity: River Crossing Vector TriangleRiver (current flows east)StartNorth bankSouth bankv_boat/water= 4 m/s Nv_water/ground= 3 m/s Ev_boat/ground= 5 m/s37° Boat vel. relative to waterCurrent vel. relative to groundResultant: boat vel. relative to ground
Origins · 1 minGalileo Galilei and how this was discovered
GG
Galileo Galilei
1564–1642 · Italian

Galileo was the first scientist to articulate the principle of relativity in mechanics — that the laws of motion are the same in any frame moving at constant velocity. In his 1632 Dialogue Concerning the Two Chief World Systems, he described a famous thought experiment set inside a ship's cabin: no mechanical experiment performed below deck could tell you whether the ship was at rest or sailing smoothly. This insight, centuries before Einstein, established the conceptual foundation for the Galilean transformations that bear his name. Galileo reportedly tested his ideas about relative motion by dropping cannonballs from the mast of a moving ship — the ball always landed at the base of the mast, not behind it, exactly as his theory predicted.

  • Displacement, Velocity & Acceleration — Relative velocity builds directly on the definition of velocity as a vector quantity; you must understand velocity before comparing velocities across frames.
  • Vector Addition and Components — Solving 2D relative velocity problems requires adding and resolving vectors into components — the core skill used in every river-crossing or wind-triangle calculation.
  • Projectile Motion — Projectile motion is often analysed by switching to the reference frame of the projectile's horizontal motion, making relative velocity ideas directly applicable.
  • Uniform Circular Motion — Centripetal acceleration involves continuously changing velocity direction; understanding reference frames helps when analysing rotating systems or orbits.