JEE Physics — Center of Mass and Collisions Complete…

Chapter Overview & Weightage

Center of Mass and Collisions is one of those chapters where JEE rewards students who understand the physics, not just the formulas. The concepts here — conservation of momentum, energy, and the elegant idea of the center of mass frame — show up across mechanics problems, sometimes disguised as “rotation” or “gravitation” questions.

Weightage: 5–7% in JEE Main | 1–2 questions in JEE Advanced

This chapter consistently delivers 1–2 questions per JEE Main paper. In JEE Advanced, COM concepts often appear embedded in multi-concept problems alongside rotation and work-energy. Never skip it — the effort-to-marks ratio is excellent.

Year	JEE Main (Questions)	JEE Advanced (Questions)	Key Topics Tested
2024	2	1 (multi-concept)	Elastic collision, COM of L-shaped body
2023	1	2	Variable mass, coefficient of restitution
2022	2	1	Impulse-momentum, inelastic collision
2021	2	1	COM of continuous body, explosion
2020	1	2	COM frame, oblique collision

The trend is clear: JEE Main tests formula application, while JEE Advanced tests the COM frame and multi-step reasoning.

Key Concepts You Must Know

Prioritised by frequency in PYQs — the ones at the top appear almost every year.

Tier 1 (Almost Guaranteed):

Conservation of linear momentum — conditions, when it applies (no external force)
Elastic vs. inelastic collision — velocity formulas, KE loss calculation
Coefficient of restitution $e$ and what $e = 0$ , $e = 1$ , $0 < e < 1$ mean physically
COM of two-particle system and system of particles

Tier 2 (High Frequency):

COM of uniform continuous bodies — rod, triangle, semicircle, hemisphere, cone
COM of composite bodies with holes (subtract-mass technique)
Impulse — its relationship to change in momentum, graphical interpretation
Explosion problems — a body at rest breaks into parts

Tier 3 (JEE Advanced / Tricky Questions):

COM frame (zero-momentum frame) — kinetic energy in COM frame vs lab frame
Oblique elastic collision — especially when one particle is at rest
Variable mass problems (rocket equation concept, not derivation)
Spring-coupled collision problems

Important Formulas

\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \cdots + m_n\vec{r}_n}{m_1 + m_2 + \cdots + m_n} = \frac{\sum m_i \vec{r}_i}{\sum m_i}

When to use: Any problem where you need the position, velocity, or acceleration of the system as a whole. If external forces are zero, $\vec{a}_{cm} = 0$ , so the COM moves at constant velocity (or stays put).

x_{cm} = \frac{\int x \, dm}{\int dm}

When to use: Uniform rods, triangular plates, semicircular wires, hemispheres. The key is setting up $dm$ in terms of $dx$ (or $d\theta$ , $dr$ ) using the uniform density assumption.

Results to memorise:

Semicircular wire: $y_{cm} = \dfrac{2R}{\pi}$ from diameter
Solid hemisphere: $y_{cm} = \dfrac{3R}{8}$ from flat face
Solid cone: $y_{cm} = \dfrac{h}{4}$ from base
Triangular lamina: $y_{cm} = \dfrac{h}{3}$ from base

v_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1 \qquad v_2 = \frac{2m_1}{m_1 + m_2} u_1

When to use: $e = 1$ , no energy loss. Special cases: $m_1 = m_2 \Rightarrow$ velocities exchange. $m_1 \gg m_2 \Rightarrow v_2 \approx 2u_1$ . $m_1 \ll m_2 \Rightarrow v_1 \approx -u_1$ (ball bounces back).

e = \frac{\text{relative speed of separation}}{\text{relative speed of approach}} = \frac{v_2 - v_1}{u_1 - u_2}

When to use: Any collision problem that gives you $e$ instead of saying “elastic” or “perfectly inelastic.” Combine this with momentum conservation to get two equations for two unknowns ( $v_1$ , $v_2$ ).

\Delta KE = \frac{1}{2} \cdot \frac{m_1 m_2}{m_1 + m_2}(u_1 - u_2)^2 (1 - e^2)

When to use: When asked “how much energy is lost?” without solving for individual velocities. This is faster than computing $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$ after finding velocities.

J = \Delta p = F_{avg} \cdot \Delta t = \int F \, dt

When to use: Problems with short-duration forces (bat hitting ball, hammer strike). The area under $F$ - $t$ graph gives impulse directly — this is a favourite graph-based question format in JEE Main.

Solved Previous Year Questions

PYQ 1 — JEE Main 2024 Shift 1

A ball of mass 0.5 kg moving at 10 m/s strikes another ball of mass 1 kg at rest. After collision, the first ball moves at 2 m/s in the same direction. Find the velocity of the second ball and check if the collision is elastic.

Step 1: Apply conservation of momentum.

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

0.5 \times 10 + 0 = 0.5 \times 2 + 1 \times v_2

5 = 1 + v_2 \implies v_2 = 4 \text{ m/s}

Step 2: Check kinetic energy.

KE_i = \frac{1}{2}(0.5)(10)^2 = 25 \text{ J}

KE_f = \frac{1}{2}(0.5)(2)^2 + \frac{1}{2}(1)(4)^2 = 1 + 8 = 9 \text{ J}

KE is not conserved, so the collision is inelastic. As a check: $e = \frac{4-2}{10-0} = \frac{2}{10} = 0.2$ .

PYQ 2 — JEE Main 2023 (COM of Composite Body)

A uniform square plate of side $a$ has a circular hole of radius $a/4$ cut at the centre. Find the COM of the remaining plate.

The trick here is the subtraction method. We treat the plate as a complete square minus the disc.

Step 1: Let the centre of the square be the origin. Both the square and the hole are centred at the origin.

x_{cm} = \frac{m_{square} \cdot 0 - m_{hole} \cdot 0}{m_{square} - m_{hole}} = 0

The COM stays at the origin because the hole is centrally placed. The answer is the centre of the plate.

If the hole is off-centre, the COM shifts away from the hole. Think of it as the remaining mass being “heavier” on the side where the hole isn’t. The formula: $x_{cm} = \frac{m_1 x_1 - m_2 x_2}{m_1 - m_2}$ where $m_2$ is the mass of the removed portion.

PYQ 3 — JEE Advanced 2022 (Explosion)

A body of mass 3 kg at rest explodes into three equal fragments. Two fragments fly off at right angles to each other with speeds 10 m/s each. Find the speed of the third fragment.

Since the body was at rest, total initial momentum = 0. So the three momentum vectors must sum to zero.

Step 1: Take the two known fragments along x and y axes.

\vec{p}_1 = 1 \times 10 \hat{i} = 10\hat{i} \text{ kg m/s}

\vec{p}_2 = 1 \times 10 \hat{j} = 10\hat{j} \text{ kg m/s}

Step 2: For momentum to be zero:

\vec{p}_3 = -(10\hat{i} + 10\hat{j})

|\vec{p}_3| = \sqrt{10^2 + 10^2} = 10\sqrt{2} \text{ kg m/s}

Step 3: Speed of third fragment (mass = 1 kg):

v_3 = \frac{10\sqrt{2}}{1} = 10\sqrt{2} \approx 14.1 \text{ m/s}

The third fragment travels at 135° to both original fragments (in the third quadrant direction).

Difficulty Distribution

Difficulty	Percentage	What to Expect
Easy	35%	Direct momentum conservation, standard elastic/inelastic formulas, COM of simple shapes
Medium	45%	Composite body COM, coefficient of restitution combined with momentum, explosion problems
Hard	20%	COM frame problems, spring-coupled collisions, oblique collisions, multi-step JEE Advanced problems

For JEE Main targeting 150+: master Easy and Medium completely. For JEE Advanced Top 1000: you need the Hard category too, especially COM frame velocity calculations.

Expert Strategy

Week 1 — Build the Foundation

Solve COM of standard shapes without looking at formulas. Derive the semicircle and hemisphere results yourself once — this locks in the integration technique for all continuous bodies.

Week 2 — Collision Mastery

Work through at least 15 collision problems in this order: elastic (1D) → inelastic (1D) → $e$ -based (1D) → oblique elastic. Most students rush to oblique collisions before mastering 1D — don’t.

The COM frame is your best friend for JEE Advanced. In the COM frame, the total momentum is zero, and after an elastic collision, particles simply reverse their COM-frame velocities. This makes oblique elastic collision solutions elegant and fast.

The PYQ Approach

This chapter’s JEE Main questions are repetitive in structure. After solving 20 PYQs, you will recognise the setup within 10 seconds. The 2019–2024 PYQ set for this chapter is your primary revision material — go through it twice.

Scoring Tip

In JEE Main, questions on COM of L-shaped or T-shaped plates are guaranteed at least once a year. The technique is always the same: split into rectangles, find each rectangle’s COM, then combine. Practise this on 5–6 different shapes and you own this question type.

Common Traps

Trap 1: Applying energy conservation in an inelastic collision

If the problem says “they stick together” or gives $e < 1$ , energy is NOT conserved. Only momentum is. Students often set up two equations (momentum + energy) and get wrong answers. Use momentum conservation + the $e$ equation as your two equations.

Trap 2: Wrong sign in the $e$ formula

$e = \frac{v_2 - v_1}{u_1 - u_2}$ — the numerator is (final velocity of the struck body minus final velocity of the striking body), and denominator is reversed. Getting the sign wrong here flips your answer. A quick sanity check: after a normal collision, relative speed of separation should be positive.

Trap 3: COM of composite body with hole — forgetting to subtract the hole’s position, not just its mass

The formula is $x_{cm} = \frac{M x_M - m x_m}{M - m}$ . Students often correctly subtract the mass but use $x_m = 0$ even when the hole is off-centre. Always locate the hole’s centroid first.

Trap 4: Momentum conservation when external forces are present

Momentum is conserved only when net external force is zero. If a block on a rough surface collides, friction from the floor is an external horizontal force — momentum is NOT conserved over the entire motion, only at the instant of collision (since collision time $\Delta t \to 0$ , the impulse from friction $\approx 0$ during impact).

Trap 5: Confusing $e = 0$ with zero velocity

$e = 0$ means perfectly inelastic — they stick together. Their combined velocity is $\frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$ , which is generally not zero. Zero final velocity only happens if the two momentum vectors cancel, which is a special case.