Collision in Physics: Complete Notes for BSc 1st Year Students (Elastic & Inelastic Collisions)
Introduction
Collision is one of the most important topics in BSc 1st Year Physics because it brings together Newton's Laws, momentum, impulse, and energy conservation into a single framework. It appears consistently in university examinations, usually as both a theory question (define and differentiate types of collision, derive formulas) and a numerical problem (calculate velocities after collision, find kinetic energy lost).
More than exam relevance, understanding collision is understanding how objects actually interact in the physical world. Every car crash, every cricket ball struck by a bat, every bullet fired from a gun, every billiard ball ricocheting off another — all of these are collisions, and they all obey the same conservation principles explored in this guide.
These notes build directly on the law of conservation of linear momentum (see the Conservation of Linear Momentum guide) and extend it to the specific scenarios called collisions. By the end, you should be able to classify any collision, apply the correct conservation equations, calculate final velocities, determine the energy lost, and solve TU-style numerical problems.
What is a Collision?
A collision is a short-duration interaction between two or more bodies during which they exert large forces on each other, causing changes in their velocities, directions, or both.
Key characteristics of a collision:
- The interaction time is short (typically milliseconds to microseconds)
- During the interaction, the internal forces between colliding bodies are very large compared to any external forces
- Because of this, total momentum of the system is conserved even if external forces (gravity, friction) technically exist — they produce negligible impulse in the short collision time
Collision vs. explosion: In a collision, bodies come together and interact. In an explosion, a single body breaks into parts. Both conserve momentum (if isolated), but they run in opposite directions in time.
Types of Collision
Collisions are classified based on what happens to kinetic energy:
| Type | Momentum Conserved | KE Conserved | Objects After |
|---|---|---|---|
| Elastic | ✅ Yes | ✅ Yes | Separate, no deformation |
| Inelastic | ✅ Yes | ❌ No (partial loss) | Separate, some deformation |
| Perfectly Inelastic | ✅ Yes | ❌ No (maximum loss) | Stick together |
A key point worth repeating: momentum is conserved in all three types. Kinetic energy is only fully conserved in elastic collisions.
1. Elastic Collision
Definition
An elastic collision is one in which both total momentum and total kinetic energy are conserved.
No real-world collision is perfectly elastic, but many are close enough that the elastic approximation gives useful answers. Billiard balls and gas molecule collisions are the standard examples.
Conditions
- No permanent deformation
- No energy converted to heat or sound
- Bodies return to their original shape after interaction
Equations for One-Dimensional Elastic Collision
Let two bodies with masses m₁ and m₂ move with initial velocities u₁ and u₂. After collision they have velocities v₁ and v₂.
Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ ... (1)
Conservation of kinetic energy: ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂² ... (2)
Derivation of Final Velocities
From equation (1): m₁(u₁ − v₁) = m₂(v₂ − u₂) ... (3)
From equation (2), cancel ½ and rearrange: m₁(u₁² − v₁²) = m₂(v₂² − u₂²) m₁(u₁ + v₁)(u₁ − v₁) = m₂(v₂ + u₂)(v₂ − u₂) ... (4)
Dividing equation (4) by equation (3): u₁ + v₁ = v₂ + u₂ u₁ − u₂ = v₂ − v₁ ... (5)
Equation (5) is important: it says the relative velocity of approach before the collision equals the relative velocity of separation after the collision, but with opposite sign.
Solving equations (3) and (5) simultaneously:
v₁ = (m₁ − m₂)u₁ + 2m₂u₂ / (m₁ + m₂)
v₂ = (m₂ − m₁)u₂ + 2m₁u₁ / (m₁ + m₂)
Special Cases
Case 1: Equal masses (m₁ = m₂ = m)
v₁ = u₂ and v₂ = u₁
The bodies exchange velocities. If m₂ is at rest (u₂ = 0): v₁ = 0, v₂ = u₁. The first body stops completely and the second moves at the first's original speed. This is why Newton's Cradle behaves the way it does.
Case 2: Very heavy body hits light body at rest (m₁ >> m₂, u₂ = 0)
v₁ ≈ u₁ (heavy body barely slows down) v₂ ≈ 2u₁ (light body flies off at nearly twice the heavy body's speed)
Case 3: Very light body hits heavy body at rest (m₁ << m₂, u₂ = 0)
v₁ ≈ −u₁ (light body bounces back at same speed) v₂ ≈ 0 (heavy body barely moves)
2. Inelastic Collision
Definition
An inelastic collision is one in which total momentum is conserved but kinetic energy is not fully conserved. Some kinetic energy is converted to heat, sound, internal deformation, or other forms.
Most real-world collisions are inelastic to some degree. A car crash, a baseball bat hitting a ball, a lump of clay being thrown at a wall — all inelastic.
Equations
Conservation of momentum still applies: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
But kinetic energy is not conserved: ½m₁u₁² + ½m₂u₂² > ½m₁v₁² + ½m₂v₂²
The difference is the kinetic energy lost: ΔKE = KE_initial − KE_final (positive value, since energy is lost)
In a general inelastic collision, you have two unknowns (v₁ and v₂) but only one conservation equation. You need additional information (like the coefficient of restitution) to solve fully.
3. Perfectly Inelastic Collision
Definition
A perfectly inelastic collision is one in which the bodies stick together after the collision and move as a single combined mass. This is the case of maximum kinetic energy loss.
Equations
After collision, both bodies move with the same final velocity v:
m₁u₁ + m₂u₂ = (m₁ + m₂)v
v = (m₁u₁ + m₂u₂) / (m₁ + m₂)
Kinetic Energy Lost
KE_initial = ½m₁u₁² + ½m₂u₂²
KE_final = ½(m₁ + m₂)v²
ΔKE = KE_initial − KE_final
This energy goes into permanent deformation, heat, and sound. In a car crash, the crumpling of the car body is the physical manifestation of this energy conversion.
4. Coefficient of Restitution
The coefficient of restitution (e) is a dimensionless number that characterizes how elastic a collision is between two specific materials.
Definition:
e = Relative velocity of separation / Relative velocity of approach
e = (v₂ − v₁) / (u₁ − u₂)
Note: this uses the convention that u₁ > u₂ (first body approaching second body), making the denominator positive.
Values of e:
| Collision Type | e Value |
|---|---|
| Perfectly elastic | e = 1 |
| Perfectly inelastic | e = 0 |
| Real inelastic collision | 0 < e < 1 |
For a ball dropped from height h onto a floor and bouncing to height h': e = √(h'/h)
The coefficient depends on the materials involved — a rubber ball on a hard floor has e ≈ 0.7–0.9; a lump of putty on a hard surface has e ≈ 0.
5. Collision in Two Dimensions
When the collision is not head-on (a glancing collision), momentum must be conserved separately in the x and y directions.
Setup: Object 1 (mass m₁, velocity u₁ along x-axis) hits stationary object 2 (mass m₂). After collision, object 1 moves at angle θ to x-axis with speed v₁, and object 2 moves at angle φ to x-axis with speed v₂.
X-direction (momentum): m₁u₁ = m₁v₁cosθ + m₂v₂cosφ
Y-direction (momentum): 0 = m₁v₁sinθ − m₂v₂sinφ
For an elastic 2D collision, add the kinetic energy equation: ½m₁u₁² = ½m₁v₁² + ½m₂v₂²
Three equations, four unknowns — typically one angle is given and you solve for the rest.
6. Impulse and Its Role in Collisions
Impulse (J) is the product of the average force during the collision and the collision duration:
J = F_avg × Î”t = Δp (change in momentum)
This connects force and momentum change. Since collision duration is very short, the force during collision must be very large to produce a significant change in momentum. This is why collisions can cause damage — large forces over short times.
Practical application: Car crumple zones and airbags increase the collision time Δt, which reduces the peak force F_avg for the same change in momentum, reducing injury.
Real-Life Applications
Newton's Cradle: Five steel balls suspended by strings. When one ball is pulled and released, it collides with the others. Due to elastic collision and equal masses, exactly one ball swings out the other side — demonstrating that momentum and kinetic energy are both transferred.
Car Crashes: The crumpling of the car body is a perfectly inelastic collision converting kinetic energy into deformation energy. Airbags extend the collision time to reduce the impact force on occupants.
Cricket/Football: A ball struck by a bat or foot undergoes an inelastic collision. The bat/foot transfers momentum to the ball. The follow-through keeps contact time longer, increasing impulse and thus ball speed.
Billiard Balls: Near-elastic collision — the balls are hard steel-coated and deform minimally. The equal-mass elastic collision case (v₁=0, v₂=u₁) explains why a cue ball stops on a straight-on shot.
Bullet and Gun: When a bullet is fired, momentum is conserved — the gun recoils backward with momentum equal and opposite to the bullet. This is an explosive separation rather than a collision, but the same conservation law applies.
Asteroid Impacts: Astronomers use collision physics to calculate the effects of asteroid impacts. The momentum transferred to Earth's crust and the energy released as heat, sound, and shock waves all follow elastic/inelastic collision equations.
Nuclear Reactions: Subatomic particle collisions in particle physics follow the same conservation laws. Elastic collision equations apply to particle scattering experiments.
Solved Numerical Problems
Problem 1 — Elastic Collision (Equal Masses)
Two balls of equal mass m = 2 kg. Ball A moves at 6 m/s and hits stationary ball B. Find the velocities after the elastic collision.
Since masses are equal and it's elastic: v₁ = 0 (ball A stops) v₂ = u₁ = 6 m/s (ball B moves at ball A's original speed)
This is the equal-mass elastic collision result — the first object stops and the second takes on all the velocity.
Problem 2 — Perfectly Inelastic Collision
A 3 kg trolley moving at 4 m/s collides with a stationary 5 kg trolley and they stick together. Find: (a) final velocity, (b) kinetic energy lost.
(a) Using (m₁ + m₂)v = m₁u₁ + m₂u₂: (3 + 5)v = 3 × 4 + 5 × 0 = 12 v = 12/8 = 1.5 m/s
(b) KE_initial = ½ × 3 × 4² = ½ × 3 × 16 = 24 J KE_final = ½ × 8 × 1.5² = ½ × 8 × 2.25 = 9 J KE lost = 24 − 9 = 15 J
Problem 3 — Elastic Collision (Different Masses)
A 4 kg ball moving at 10 m/s collides elastically with a 2 kg stationary ball. Find the velocity of both balls after collision.
Using the elastic collision velocity formulas:
v₁ = (m₁ − m₂)u₁/(m₁ + m₂) = (4 − 2)(10)/(4 + 2) = 20/6 = 3.33 m/s (forward)
v₂ = 2m₁u₁/(m₁ + m₂) = 2 × 4 × 10/(4 + 2) = 80/6 = 13.33 m/s (forward)
Verification (momentum): 4×10 = 40; 4×3.33 + 2×13.33 = 13.33 + 26.67 = 40 ✓ Verification (KE): ½×4×100 = 200 J; ½×4×11.11 + ½×2×177.67 = 22.22 + 177.67 ≈ 200 J ✓
Problem 4 — Coefficient of Restitution
A ball of mass 0.5 kg is dropped from a height of 2 m and bounces to a height of 1.25 m. Find the coefficient of restitution.
e = √(h'/h) = √(1.25/2) = √(0.625) = 0.79
This ball retains 79% of its separation speed relative to its approach speed — a reasonably elastic ball (like a good rubber ball on a hard floor).
Problem 5 — Inelastic Collision with Energy Calculation
A 6 kg body moving at 3 m/s collides with a 4 kg body moving at 1 m/s in the same direction. If e = 0.5, find the velocities after collision.
Step 1 — Conservation of momentum: 6×3 + 4×1 = 6v₁ + 4v₂ 18 + 4 = 6v₁ + 4v₂ 6v₁ + 4v₂ = 22 ... (i)
Step 2 — Coefficient of restitution: e = (v₂ − v₁)/(u₁ − u₂) = 0.5 (v₂ − v₁) = 0.5(3 − 1) = 1 v₂ − v₁ = 1 ... (ii)
Step 3 — Solve simultaneously: From (ii): v₂ = v₁ + 1 Substitute into (i): 6v₁ + 4(v₁ + 1) = 22 10v₁ = 18 v₁ = 1.8 m/s v₂ = 2.8 m/s
Formula Quick-Reference
| Formula | Description |
|---|---|
| m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | Conservation of momentum (all collisions) |
| ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂² | Conservation of KE (elastic only) |
| v₁ = [(m₁−m₂)u₁ + 2m₂u₂]/(m₁+m₂) | Velocity of body 1 after elastic collision |
| v₂ = [(m₂−m₁)u₂ + 2m₁u₁]/(m₁+m₂) | Velocity of body 2 after elastic collision |
| v = (m₁u₁+m₂u₂)/(m₁+m₂) | Perfectly inelastic: common final velocity |
| e = (v₂−v₁)/(u₁−u₂) | Coefficient of restitution |
| e = √(h'/h) | e from bounce heights |
| e = 1 (elastic), e = 0 (perfectly inelastic) | Limiting values |
| ΔKE = KE_initial − KE_final | Kinetic energy lost |
Important Exam Questions
Long Answer (8–10 Marks)
- Define elastic and inelastic collision. Derive expressions for the velocities of two bodies after a one-dimensional elastic collision.
- What is a perfectly inelastic collision? Derive an expression for the final velocity and the kinetic energy lost.
- Define coefficient of restitution. Explain its significance and derive its relation to bounce height.
- Explain two-dimensional collision. Write the conservation equations for x and y directions.
- State and prove the law of conservation of momentum. Apply it to a collision problem.
Short Answer (3–5 Marks)
- Differentiate between elastic and inelastic collisions with two examples each.
- What is coefficient of restitution? What are its values for elastic and perfectly inelastic collisions?
- Explain what happens in a head-on elastic collision between equal masses.
- Define impulse. How does it relate to collision?
- Why is kinetic energy not conserved in an inelastic collision?
- Give three real-life examples of perfectly inelastic collisions.
Numerical Questions
- A 2 kg ball at 5 m/s elastically collides with a 2 kg stationary ball. Find final velocities.
- A 5 kg object at 8 m/s hits a stationary 3 kg object and they stick together. Find final velocity and KE lost.
- A ball is dropped from 3 m and bounces to 1.5 m. Find the coefficient of restitution.
Past Question Trend Analysis
Appears almost every year:
- Define collision. Differentiate elastic and inelastic collision
- Derive velocities after one-dimensional elastic collision
- Coefficient of restitution — definition and formula
- Numerical on perfectly inelastic collision (final velocity + KE lost)
Frequently appears:
- Perfectly inelastic collision derivation
- Two-dimensional collision equations
- Numerical using coefficient of restitution
- Real-life applications of collision
Occasionally appears:
- Special cases of elastic collision (equal masses, m₁>>m₂)
- Impulse in the context of collisions
Frequently Asked Questions
What is a collision in physics? A collision is a short-duration interaction between two or more bodies in which large mutual forces cause changes in velocity. Total momentum is conserved in all collisions (in an isolated system).
What are the types of collisions? Elastic (both momentum and KE conserved), inelastic (momentum conserved, KE partially lost), and perfectly inelastic (momentum conserved, maximum KE lost, bodies stick together).
Is momentum always conserved in a collision? Yes — provided no net external force acts on the system. In practice, collision times are so short that external forces (gravity, friction) produce negligible impulse, so momentum conservation is a reliable tool.
Is kinetic energy always conserved in a collision? No. Only elastic collisions conserve kinetic energy. In all real collisions, some KE is converted to heat, sound, deformation energy, or other forms.
What does coefficient of restitution measure? It measures how "bouncy" a collision is — the ratio of relative separation speed to relative approach speed. e = 1 for perfectly elastic, e = 0 for perfectly inelastic.
What is the difference between collision and explosion? In a collision, two bodies approach and interact. In an explosion, one body breaks into parts that fly apart. Both conserve total momentum, but in an explosion, chemical or internal energy is converted to kinetic energy (KE increases), whereas in an inelastic collision, KE decreases.
Why do crumple zones reduce injury in car crashes? Crumple zones increase the time over which the deceleration happens (longer Δt). Since impulse = FΔt = constant (same momentum change), a longer Δt means a smaller peak force F on the occupants, reducing injury severity.
What is the SI unit of impulse? Newton-second (N·s), which is equivalent to kg·m/s — the same unit as momentum.
Conclusion
Collision is a topic that unifies momentum, energy, and Newton's Laws into a single framework with direct real-world relevance. The three collision types — elastic, inelastic, and perfectly inelastic — cover every interaction between objects in classical mechanics, from subatomic particles to automobile crashes.
For exams, the most reliable preparation strategy is: memorize the definitions precisely, understand the derivation of elastic collision velocity formulas (it's a two-equation system that appears every year), and practice numerical problems covering all three types including the coefficient of restitution.
For related physics topics: Conservation of Linear Momentum, Newton's Laws of Motion — Complete Guide, Kinematics: Complete Notes for BSc 1st Year Physics, and BSc CSIT Physics (PHY118) Complete Guide.
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