Lesson 6 – Centre of Mass & System of Particles
IIT–JEE Tough Problems
Part 1 – Long & High-Difficulty Numericals
This section contains genuinely tough IIT–JEE problems. These questions demand patience, clean logic, and system-level thinking.
Problem 1 (IIT–Advanced · Multi-Step)
A shell of mass 3 kg is moving horizontally with velocity 12 m/s. It explodes into two fragments of masses 1 kg and 2 kg. The lighter fragment moves vertically upward with velocity 6 m/s. Find the velocity of the heavier fragment immediately after explosion.
Solution:
Initial momentum (horizontal) = 3 × 12 = 36 kg·m/s
Momentum must be conserved in both x and y directions.
Vertical direction:
Initial vertical momentum = 0
Final: 1×6 + 2×vy = 0
vy = −3 m/s
Horizontal direction:
1×0 + 2×vx = 36
vx = 18 m/s
Answer:
Velocity of heavier fragment =
(18 î − 3 ĵ) m/s
Problem 2 (IIT–Advanced · COM Motion)
A man of mass m walks from one end to the other end of a boat of mass 2m and length L floating in still water. Find the displacement of the boat relative to the shore.
Solution:
No external horizontal force acts on the system.
Hence, centre of mass remains fixed.
Let boat shift by distance x opposite to man’s motion.
m(L − x) + 2m(−x) = 0
mL − mx − 2mx = 0
3mx = mL
x = L / 3
Answer:
Boat moves backward by L/3.
Problem 3 (IIT–Advanced · Constraint + COM)
Two blocks of masses m and 2m are connected by a light spring and placed on a smooth horizontal surface. The spring is compressed and released. Find the velocity of centre of mass during motion.
Solution:
Spring force is internal to the system.
No external horizontal force acts.
Hence, velocity of centre of mass remains constant.
If system was initially at rest, COM velocity remains zero throughout.
Answer:
Velocity of centre of mass = 0.
Problem 4 (IIT–Advanced · Variable Mass Insight)
A block of ice slides on a frictionless surface with velocity v. It melts uniformly. Find the velocity of the centre of mass of the system.
Solution:
Melting is an internal process.
No external horizontal force acts.
Hence, centre of mass continues to move with same velocity v.
Answer:
Velocity of COM = v.
Problem 5 (IIT–Advanced · Conceptual Killer)
Can the centre of mass of a system move even when all external forces are zero? Explain with an example.
Solution:
Yes, if initial momentum of the system is non-zero.
Example: two particles moving with same velocity in same direction.
Even without external force, COM continues uniform motion.
Answer:
Yes, COM can move with constant velocity.
TOUGH PROBLEM SOLVING STRATEGY (Part–1)
✔ Always conserve momentum in x and y separately
✔ Explosion → solve components independently
✔ Identify system boundary clearly
✔ No external force ⇒ COM velocity constant
✔ Never rush — these problems reward patience
📘 Centre of Mass & System of Particles – Complete Physics Library
This library is a complete learning package for Intermediate + IIT JEE Physics covering Centre of Mass and System of Particles.
It includes theory, solved examples, objective questions, IIT-level problems, revision strategies, tricks, and exam-oriented guidance.
📂 Stage 1: Intermediate Complete Notes (Concept Foundation)
📂 Stage 2: Advanced Concepts & Applications
📂 Stage 3: Objective Questions, Tough Problems & Revision
🔗 Learning Hub & Community
Prepared by: Shaktimatha Learning 🌱
Vision: Concept clarity • Logical thinking • Exam excellence
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