Q.Two cars move towards each other along the same straight road, one with a velocity of 20 m/s and the other with a velocity of 15 m/s. Find the velocity with which the two cars approach each other.
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Imagine you're sitting in a train that's moving smoothly. The person sitting opposite you appears to be perfectly still — yet both of you are hurtling past trees and buildings outside at 80 km/h. Which is the "real" velocity? The answer is: there is no single real velocity. Velocity always depends on who is measuring it.
That's the core idea of relative velocity: the velocity of an object as seen from a particular frame of reference. Change the frame, and the measured velocity changes.
The Intuition: Walking on a Moving Train
Let's build this step by step.
Step 1 — You on a stationary train.
You walk forward at 3 km/h inside the aisle. A friend on the platform sees you moving at exactly 3 km/h. Simple.
Step 2 — The train moves at 80 km/h, you stand still inside.
Your friend on the platform sees you moving at 80 km/h (the train's speed). You see the platform rushing backward at 80 km/h.
Step 3 — You walk forward at 3 km/h while the train moves at 80 km/h.
Your friend on the platform sees you moving at 80+3=83 km/h.
But the person sitting next to you sees you moving at just 3 km/h.
Same you, same walking speed — two different observers, two different velocities. That's relative velocity in action.
Note
The "velocity" you feel is always relative to something. When you say "a car is moving at 60 km/h", you usually mean relative to the ground. But the ground itself is moving (Earth rotates, orbits the Sun, etc.). There is no absolute rest frame.
The Precise Definition
Relative velocity of object A with respect to object B is the velocity of A as measured by an observer who is at rest with respect to B.
Mathematically, if vA and vB are velocities of A and B measured in the same frame (say, the ground), then:
vAB=vA−vB
Where vAB means "velocity of A relative to B".
Read this carefully: you subtract the velocity of the reference object (B) from the velocity of the object you're tracking (A).
Why Subtraction? — The Logic
Think of the train example again. Let:
vyou = your velocity relative to ground = 83 km/h forward
vtrain = train's velocity relative to ground = 80 km/h forward
Your velocity relative to the train is:
vyou,train=vyou−vtrain=83−80=3 km/h forward
That matches: the person on the train sees you walking forward at 3 km/h.
Now what about the platform's velocity relative to you?
Platform is at rest relative to ground: vplatform=0
vplatform, you=0−83=−83 km/h
The negative sign means the platform appears to move backward relative to you — which is exactly what you see from the moving train.
Watch out
A common mistake: thinking relative velocity is just adding speeds. It's vector subtraction. If two objects move in opposite directions, you subtract a negative — which becomes addition. Always use the vector formula.
One-Dimensional Cases (The Simplest)
When motion is along a straight line, we can use signs (+ for one direction, − for the opposite).
Setting up. Take the direction of motion of the first car as positive. Then car A has velocity vA=+20 m/s, and car B, moving toward A along the same line (i.e. in the opposite direction), has velocity vB=−15 m/s.
Relative velocity of A with respect to B. Using the definition from Section 2.9,
vAB=vA−vB=20−(−15)=20+15=35m/s
Physical interpretation. The magnitude of this relative velocity, 35 m/s, is the rate at which the distance separating the two cars is closing — i.e. the rate of approach. Notice this is the SUM of the two individual speeds (20 m/s and 15 m/s), not their difference, precisely because the cars move in opposite directions: subtracting a negative velocity is the same as adding its magnitude. This is a general feature — whenever two bodies on the same straight lin …