Velocity of Centre of Mass When Particles Move Under Mutual Force
Physics · System Of Particles And Rotational Motion · NEET
The velocity of the centre of mass is V_cm = (m1v1 + m2v2)/M. When two particles move only under their mutual (internal) force, no external force acts, so V_cm never changes. If they both start from rest, V_cm stays zero forever, no matter how fast each particle moves. Memory hook: internal forces are a "family fight" — they push each other around, but the family's centre never moves.
Both particles start from rest and pull each other in. The lighter B moves faster (3v) than the heavier A (v), so m1:m2 = 3:1. Their momenta are equal and opposite, so the centre of mass (green line) never moves: V_cm = 0.
Your doubts, answered
If both particles are moving, how can the centre of mass stay still?
Because the two velocities point in opposite directions and their momenta cancel. V_cm = (m1v1 + m2v2)/M uses velocity as a vector (with sign). When A moves right with momentum p and B moves left with momentum p, m1v1 + m2v2 = 0, so V_cm = 0. The particles move, but the balance point between them does not.
Why is the mutual attraction not counted as a force on the system?
The attraction is an internal force. A pulls B and B pulls A with equal and opposite forces (Newton's third law). When you add all forces on the whole system, these two cancel out. Only external forces (gravity from outside, friction, applied push) can change V_cm. With no external force, F_ext = 0, so acceleration of the centre of mass A_cm = 0.
Both start at rest, then A has speed v and B has speed 3v. What is V_cm?
Zero. They started at rest, so V_cm was 0. Since only mutual (internal) force acts, F_ext = 0 and V_cm cannot change. So even when A moves at v and B at 3v, V_cm is still 0. This is the exact NEET 2023 question — the answer is zero, not v, 2v or 4v. (The 3v also tells you m1:m2 = 3:1, since m1v = m2(3v).)
What is the difference between the velocity of a particle and the velocity of the centre of mass?
A particle's velocity is how fast that one point moves. The centre of mass velocity is the mass-weighted average of all particle velocities: V_cm = (m1v1 + m2v2)/M. Individual particles can speed up, slow down or reverse due to internal forces, while V_cm depends only on total momentum P = MV_cm, which internal forces never change.
Does this mean momentum is conserved?
Yes. Since P = MV_cm and V_cm is constant when F_ext = 0, the total linear momentum P is conserved. Constant velocity of centre of mass and conservation of total momentum are two ways of saying the same thing.
⚠️ The NEET trap ✗ Students see A moving at v and B moving at 3v, add the speeds, and pick v, 2v or 4v as the centre of mass speed. ✓ With no external force and both starting from rest, V_cm stays exactly zero. Speeds are added as vectors (opposite signs), and the momenta cancel: m1v + m2(-3v) = 0. 🧠 No external force + started at rest = V_cm is zero, always. Do not add the speeds — check the forces first.
Real NEET questions
NEET 2023 Phase 2
Two particles A and B, initially at rest, move towards each other under their mutual attraction. When A's speed is v and B's speed is 3v, the speed of the centre of mass of the system is:
A · v
B · 4v
C · 2v
D · zero ✓
Solution: Step 1: Identify the forces. The only force is the mutual attraction between A and B. This is an internal force (Newton's third law pair), so the net external force F_ext = 0.
Step 2: Apply the rule for centre of mass. When F_ext = 0, the acceleration of the centre of mass A_cm = 0, so V_cm is constant (it cannot change).
Step 3: Use the initial condition. Both particles start from rest, so initially V_cm = (m1(0) + m2(0))/M = 0.
Step 4: Conclude. Since V_cm was 0 and cannot change, V_cm stays 0 at every later instant — even when A moves at v and B at 3v. (Check: momentum conservation gives m1v = m2(3v), so m1:m2 = 3:1, and m1v - m2(3v) = 0.)
Answer: zero (option D).
Solved System Of Particles And Rotational Motion NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
