Relative Velocity – University‑Level Study Notes
Physics – Kinematics
Prepared for IIT‑JEE (Main + Advanced) – MIT‑style depth
1. Introduction
Relative velocity is the velocity of one particle as seen by another particle.
For two particles (A) and (B) moving in an inertial frame (S) with position vectors
(\mathbf{r}_A(t)) and (\mathbf{r}_B(t)), the relative position is
[ \boxed{\mathbf{r}_{BA}(t)=\mathbf{r}_B(t)-\mathbf{r}_A(t)} ]
The relative velocity is the time derivative:
[ \boxed{\mathbf{v}{BA}(t)=\frac{d\mathbf{r}{BA}}{dt} =\mathbf{v}_B(t)-\mathbf{v}_A(t)} ]
where (\mathbf{v}_i(t)=d\mathbf{r}_i/dt) is the velocity of particle (i) in the inertial frame (S).
This relation is valid for any coordinate system that is linear (Cartesian) and any differentiable motion.
2. Derivation from First Principles
2.1 One‑Dimensional Motion
Let (x_A(t)) and (x_B(t)) be the positions in a 1‑D inertial frame.
Define the distance between them as (d(t)=x_B(t)-x_A(t)).
Differentiating:
[ \frac{dd}{dt} = \frac{dx_B}{dt} - \frac{dx_A}{dt} = v_B - v_A. ]
Hence
[ v_{BA} = \frac{dd}{dt} = v_B - v_A . ]
Interpretation: If (B) moves faster than (A) ((v_B>v_A)), (d) increases; if (v_B<v_A), (d) decreases.
2.2 Multi‑Dimensional Motion
In vectors, the same derivative applies:
[ \mathbf{v}_{BA} = \frac{d}{dt}(\mathbf{r}_B-\mathbf{r}_A) = \mathbf{v}_B - \mathbf{v}_A . ]
The magnitude (|\mathbf{v}_{BA}|) is the relative speed, but direction matters for most applications.
2.3 Relative Velocity in Non‑Inertial Frames
If observer (A) is accelerating, i.e. in a non‑inertial frame, the relative velocity acquires an extra term when expressed in (A)’s coordinate system.
Let (S') be the frame attached to (A). Then
[ \boxed{\mathbf{v}{BA}^{(S')} = \mathbf{v}{BA}^{(S)} - \mathbf{v}_{AA}^{(S)}} ]
where (\mathbf{v}_{AA}^{(S)}) is the velocity of (A) in the inertial frame (S).
If (A) is rotating, we must also account for Coriolis and centripetal terms in the relative acceleration, but the velocity transformation remains a simple Galilean shift.
3. Vector Addition of Velocities
3.1 Galilean Relativity
In classical mechanics, the relative velocity is invariant under a Galilean transformation:
[ \mathbf{v}' = \mathbf{v} - \mathbf{V} ]
where (\mathbf{V}) is the relative velocity between the two inertial frames.
Thus, when two observers (A) and (B) are in inertial frames moving with constant relative velocity (\mathbf{V}), the velocity of (B) as seen by (A) is
[ \mathbf{v}_{BA} = \mathbf{v}_B - \mathbf{v}_A = \mathbf{v}_B' . ]
3.2 Relativistic Velocity Addition (Optional for JEE Advanced)
When speeds approach (c), Galilean addition fails. The Lorentz transformation yields
[ \mathbf{u}' = \frac{\mathbf{u} + \left[\left(\gamma-1\right)\frac{\mathbf{u}\cdot\mathbf{V}}{V^2} - \gamma\right]\mathbf{V}}{\gamma\left(1-\frac{\mathbf{u}\cdot\mathbf{V}}{c^2}\right)} ]
with (\gamma = 1/\sqrt{1-\frac{V^2}{c^2}}).
For 1‑D motion this reduces to
[ u' = \frac{u+V}{1+\frac{uV}{c^2}} . ]
This formula is rarely used in the IIT‑JEE syllabus but is essential for advanced study.
4. Deeper Theory and Edge Cases
| Edge Case | Statement | Discussion |
|---|---|---|
| Zero Relative Velocity | (v_{BA}=0) iff (v_A = v_B) | Particles are co‑moving. Relative position remains constant if initially zero. |
| Infinite Relative Velocity | Not physically realizable; would imply instantaneous separation change, violating finite propagation speed. | |
| Relative Acceleration | (\mathbf{a}_{BA} = \mathbf{a}_B - \mathbf{a}_A) | If (A) accelerates, the relative velocity changes even if (B) is inertial. |
| Non‑inertial Observer | (\mathbf{v}{BA}^{(S')} = \mathbf{v}{BA}^{(S)} - \mathbf{v}_{AA}^{(S)}) | Additional terms arise for rotating frames: Coriolis, centrifugal. |
| Relative Speed vs Relative Velocity | Speed is magnitude; velocity is vector. | Direction matters in collision |