Relative Motion
Relative motion is the motion of an object as observed from a particular reference frame, which may itself be moving. The velocity of an object relative to a second observer equals the object's velocity minus the observer's velocity (in classical mechanics). This principle explains why objects appear to move differently to observers in different states of motion, and it forms the foundation of Galilean relativity.
Key Formula
v(A relative to C) = v(A relative to B) + v(B relative to C)
LaTeX: \vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}
| Symbol | Meaning | Unit |
|---|---|---|
| v⃗_{A/C} | Velocity of A as seen from C | m/s |
| v⃗_{A/B} | Velocity of A as seen from B | m/s |
| v⃗_{B/C} | Velocity of B as seen from C | m/s |
Worked Example
Problem
Train A moves east at 60 km/h relative to the ground. Train B moves east at 40 km/h relative to the ground on a parallel track. What is the velocity of Train A as seen by a passenger on Train B?
Solution
Step 1: Let ground = reference frame C, Train B = observer (frame B), Train A = object (A). Step 2: v(A/C) = +60 km/h (east); v(B/C) = +40 km/h (east). Step 3: v(A/B) = v(A/C) − v(B/C) = 60 − 40 = 20 km/h east.
Answer
Train A appears to move at 20 km/h east relative to Train B's passenger.
Examples of relative motion in everyday situations
| Situation | Observer | Object | Relative velocity |
|---|---|---|---|
| Two trains moving same direction | Passenger on slower train | Faster train | Difference of speeds |
| Two trains moving opposite directions | Passenger on one train | Other train | Sum of speeds |
| River and swimmer | Bank (stationary) | Swimmer crossing river | Vector sum of swim + current |
| Aircraft in wind | Ground observer | Aircraft | Vector sum of airspeed + wind |
| Person walking in moving bus | Road observer | Person | Vector sum of walking + bus speed |
| Earth and Moon | Sun (approx. inertial) | Moon | Orbital velocity − Earth orbital velocity |
Interactive Tools
PhET My Solar System Simulation
Observe relative motion between orbiting bodies from different reference frames
Open ToolGeoGebra Relative Velocity
Interactive vector addition tool for solving relative velocity problems
Open ToolKhan Academy — Relative Motion
Introduction to reference frames and relative velocity with river and boat examples
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Velocity
Velocity is the rate of change of displacement with respect to time, making it a vector quantity with both magnitude (speed) and direction. Average velocity equals total displacement divided by total time, while instantaneous velocity is the derivative of position with respect to time. Velocity is central to Newton's laws and is measured in metres per second (m/s).
Displacement
Displacement is the shortest straight-line distance between an object's initial and final positions, measured as a vector quantity with both magnitude and direction. Unlike distance, displacement does not account for the actual path taken, only the net change in position. It is the fundamental quantity used to define velocity and is measured in metres (m).
Projectile Motion
Projectile motion is the two-dimensional motion of an object launched into the air that moves under the influence of gravity alone, following a curved (parabolic) trajectory. The horizontal and vertical components of motion are independent: the horizontal component is uniform (constant velocity), while the vertical component is uniformly accelerated by gravity (g ≈ 9.8 m/s²). Galileo first described this decomposition in the early 17th century.
From Latin "relativus" (having reference to something), from "relatus" (carried back), past participle of "referre" (to bring back, relate). Galileo formalised the concept of relative motion in "Dialogue Concerning the Two Chief World Systems" (1632).