Collisions represent the final and one of the most conceptually powerful sections of Chapter 5. This topic brings together all the major ideas discussed so far — work, energy, momentum, impulse, and conservation laws — into a unified analytical framework.
In competitive examinations like JEE Main and JEE Advanced, collision problems are rarely straightforward. They are often multi-step, combined with springs, wedges, inclined planes, pulleys, or rotational motion. Therefore, mastering collisions requires both conceptual clarity and mathematical precision.
At Deeksha Vedantu, we guide students to understand collisions not merely as formula-based problems but as conservation-law-driven physical processes.
What is a Collision?
A collision occurs when two bodies interact with each other over a very short time interval and exert large forces on each other.
Key features of collisions:
- Interaction time is extremely small.
- Forces involved are very large (impulsive forces).
- External forces during collision are usually negligible.
- Linear momentum conservation becomes the central governing principle.
Even though forces are large, the short time interval ensures that impulse remains finite.
Impulse–Momentum Theorem
Impulse is defined as force multiplied by time interval:
From Newton's Second Law:
Integrating over time:
Thus:
Impulse equals change in momentum.
This theorem is the foundation of all collision analysis.
Law of Conservation of Linear Momentum
If no external force acts on a system, total linear momentum remains conserved.
For two bodies of mass 
and 
:
Momentum conservation holds for:
- Elastic collisions
- Inelastic collisions
- Perfectly inelastic collisions
Momentum is always conserved in isolated systems.
Types of Collisions
1. Elastic Collision
Characteristics:
- Momentum conserved
- Kinetic energy conserved
Elastic collisions occur ideally between gas molecules and in idealized mechanical systems.
2. Inelastic Collision
- Momentum conserved
- Kinetic energy not conserved
Some kinetic energy converts into heat, sound, deformation, or internal energy.
3. Perfectly Inelastic Collision
Objects stick together after collision.
Momentum equation:
This case results in maximum kinetic energy loss.
One-Dimensional Elastic Collision – Complete Mathematical Derivation
Using momentum conservation:
Using kinetic energy conservation:
Solving these simultaneously gives:
These formulas are extremely important in JEE Advanced and often tested in conceptual variations.
Special Cases of Elastic Collision
Equal Masses
If 
and 
:
The moving particle transfers its entire velocity.
Heavy Target (
)
)The lighter particle rebounds nearly with the same speed.
Light Target ()
)The heavy particle continues almost unaffected.
Understanding limiting cases improves conceptual clarity.
Coefficient of Restitution
Defined as ratio of relative speed of separation to relative speed of approach:
Values of 
:
→ Elastic collision
→ Perfectly inelastic collision
→ Partially elastic collision
→ Elastic collision
→ Perfectly inelastic collision
→ Partially elastic collisionUsing restitution along with momentum conservation helps solve partially elastic collision problems.
Kinetic Energy Loss in Perfectly Inelastic Collision
Loss of kinetic energy:
This formula shows energy loss depends on relative velocity.
Collision in Center-of-Mass (CM) Frame
Velocity of center of mass:
In CM frame:
- Total momentum is zero.
- In elastic collision, velocities reverse direction.
Mathematically:
CM frame simplifies advanced JEE problems.
Two-Dimensional Collisions
Momentum conservation applies independently in each direction.
Energy conserved only in elastic collisions.
Example Insight
If one particle initially at rest and collision elastic, final velocities are perpendicular in lab frame (for equal masses).
This is a famous JEE concept.
Advanced JEE Problem 1 (Elastic Collision with Stationary Target)
A particle of mass 
moving with velocity 
collides elastically with stationary mass 
.
Students must:
- Apply momentum conservation.
- Apply kinetic energy conservation.
- Interpret the physical meaning of final velocities.
Advanced JEE Problem 2 (Perfectly Inelastic + Spring System)
Two blocks collide and stick together, then compress a spring.
Step 1: Momentum conservation
Step 2: Energy conservation during compression
This integrates collision with spring energy concepts.
Advanced JEE Problem 3 (Block–Wedge Collision)
If the wedge is free to move, momentum is conserved horizontally.
Energy conserved if collision elastic.
Such problems test system isolation and frame understanding.
Advanced JEE Problem 4 (Explosion as Reverse Collision)
If object at rest explodes into two fragments:
Explosion problems also follow momentum conservation.
Common Conceptual Mistakes
- Assuming kinetic energy conserved in all collisions.
- Ignoring sign conventions.
- Forgetting momentum conservation in each direction.
- Mixing up restitution formula.
- Neglecting system isolation.
Comprehensive Formula Table
| Concept | Formula |
| Momentum |  |
| Impulse | |
| Momentum Conservation | |
| Coefficient of Restitution | |
| KE Loss (Perfectly Inelastic) |  |
| CM Velocity | |
FAQs
Q1. Is momentum always conserved in collisions?
Yes, provided the system is isolated and external force is negligible.
Q2. Is kinetic energy conserved in all collisions?
No. It is conserved only in elastic collisions.
Q3. What does coefficient of restitution measure?
It measures elasticity of collision and relative speed change.
Q4. Why is center-of-mass frame useful?
Because collision equations simplify significantly in the CM frame.
Q5. What is the key strategy in solving collision problems?
Apply momentum conservation first. Then apply energy conservation if collision is elastic.
Conclusion
Collisions represent the culmination of work–energy–momentum principles. Momentum conservation governs all collisions, while energy conservation distinguishes elastic from inelastic interactions.
Mastering collisions allows students to confidently solve complex, multi-stage mechanics problems in board examinations and competitive exams like JEE Main and JEE Advanced.
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