Step-by-Step Derivation of Final Velocities in 1D Elastic Collision
Elastic Collisions In One Dimension is a central Physics topic where two bodies collide and both momentum and kinetic energy are conserved. This concept is crucial for JEE Main, as it demonstrates how fundamental laws govern everyday collisions, from carts on rails to atomic particles. Students often encounter both numerical problems and derivations based on this topic in exams.
In such collisions, objects move only along a single straight line. The interaction is called "elastic" because there is no loss of kinetic energy during the collision process—unlike real-world impacts that often result in deformation or heat generation. For rigorous JEE preparation, mastering the equations and understanding the step-by-step derivation is essential.
Key Principles in Elastic Collisions In One Dimension
Analyzing elastic collisions requires an understanding of two core laws: the conservation of linear momentum and the conservation of kinetic energy. Both are applied simultaneously to solve for final velocities and analyze motion before and after collision.
- Momentum: p = m v (where m is mass, v is velocity)
- Total initial momentum equals total final momentum.
- Kinetic energy: KE = ½ m v2
- Total initial kinetic energy equals total final kinetic energy.
- Only true for ideal cases—real collisions may lose energy.
For example, two trolleys colliding on a frictionless track showcase these rules perfectly. Such clear cases are often the ones tested in JEE Main problems.
Step-By-Step Derivation for Elastic Collisions In One Dimension
Derivation is a must-know for JEE aspirants. It involves expressing final velocities (v1f, v2f) in terms of initial velocities and masses.
- Let bodies 1 and 2 have masses m1, m2 and initial velocities u1, u2.
- Apply conservation of momentum:
m1u1 + m2u2 = m1v1f + m2v2f - Apply conservation of kinetic energy:
½ m1u12 + ½ m2u22 = ½ m1v1f2 + ½ m2v2f2 - Solve the above equations to find:
v1f = [(m1 - m2)u1 + 2m2u2] / (m1 + m2) - v2f = [(m2 - m1)u2 + 2m1u1] / (m1 + m2)
Careful substitution, clean algebra, and using the correct sign convention are critical steps—many errors occur here. Practice helps avoid missing minus signs or mixing up initial and final velocities. For additional practice, refer to collision topic and related question sets.
Comparison Table: Elastic Collisions In One Dimension vs Two Dimensions
| Aspect | 1D Elastic Collision | 2D Elastic Collision |
|---|---|---|
| Direction | Single straight line | Multiple axes/planes |
| Equations Needed | Two equations (momentum, kinetic energy) | Three or more equations (vector components) |
| JEE Main Frequency | Very common | Less common, advanced |
| Example | Balls on a straight track | Billiards, atomic scattering |
If you want a detailed two-dimensional approach, visit elastic collision in two dimensions for vector-based problems.
Numerical Example: Elastic Collisions In One Dimension Application
A body of mass 2 kg moving at 3 m/s collides elastically with a stationary body of mass 3 kg. Find final velocities.
- Given: m1 = 2 kg, u1 = 3 m/s
- m2 = 3 kg, u2 = 0 m/s
- Apply formula:
v1f = [(2-3)×3 + 2×3×0]/(2+3) = -0.6 m/s
v2f = [(3-2)×0 + 2×2×3]/(2+3) = 2.4 m/s - Final result: First body moves at -0.6 m/s (reverses), second at 2.4 m/s.
This problem structure mirrors those in JEE Main practice tests and is ideal for solidifying the formula’s application.
Revise foundational concepts about conserved quantities at conservation of momentum and sharpen your calculations further with coefficient of restitution models. Visit laws of motion for background essentials.
Common Pitfalls and Practical Tips on Elastic Collisions In One Dimension
- Confusing elastic with inelastic collision: check if kinetic energy is conserved.
- Forgetting to use sign convention—velocities to the left are negative.
- Mishandling equal mass case; velocities swap in such collisions.
- Assuming real objects are perfectly elastic. In reality, most lose some energy.
- Missing proper algebraic steps in derivation; always write momentum and kinetic energy equations first.
Explore the subtle differences with oblique collisions, or practice with more varieties in motion in one dimension modules.
Further Applications and Interlinked Topics for Elastic Collisions In One Dimension
Understanding elastic collisions boosts your performance in waves, molecular collisions, and kinetic theory. They are foundational for more complex chapters in rotational and translational motion.
- work energy and power are built on collision analysis.
- impulse momentum theorem is a helpful shortcut.
- Problems in center of mass and motion use similar logic.
- For conceptual mastery, refer to laws of motion practice paper and revision notes.
Vedantu’s JEE Physics resources offer detailed notes, mock tests, and clear stepwise solutions for elastic collision topics. By using the correct approach and practicing consistently, you’ll master even the trickiest exam questions on this theme.
FAQs on Elastic Collisions in One Dimension: Concepts, Derivation, and Examples
1. What is an elastic collision in one dimension?
An elastic collision in one dimension is a collision between two objects moving along a straight line in which both momentum and kinetic energy are conserved. In such collisions, the objects bounce off each other without any loss of total kinetic energy. Common examples include:
- Collisions between billiard balls on a straight path
- Idealized gas molecule interactions
- Head-on car crashes (in theory)
This concept is a key topic in Class 11 Physics, especially for students preparing for JEE and board exams.
2. How is kinetic energy conserved in a 1D elastic collision?
Kinetic energy is conserved in a one-dimensional elastic collision because no energy is lost to heat, sound, or deformation. The sum of the kinetic energies of both objects before and after the collision remains the same. The conservation is described by:
- Total kinetic energy before = Total kinetic energy after
- 1/2 m₁u₁² + 1/2 m₂u₂² = 1/2 m₁v₁² + 1/2 m₂v₂²
This principle is fundamental to solving numerical problems for JEE Main and Class 11 physics.
3. How do you derive the final velocities after a 1D elastic collision?
To derive the final velocities (v₁ and v₂) after a 1D elastic collision, you use both the conservation of momentum and conservation of kinetic energy equations. The steps are:
- Write the momentum conservation equation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Write the kinetic energy conservation equation: 1/2 m₁u₁² + 1/2 m₂u₂² = 1/2 m₁v₁² + 1/2 m₂v₂²
- Solve these two equations simultaneously for v₁ and v₂
This derivation is often asked in exams and forms the basis for solving related numerical problems.
4. What is the formula for final velocity in 1D elastic collision?
The final velocities of two objects after a 1D elastic collision are given by:
- v₁ = [(m₁ - m₂)u₁ + 2m₂u₂] / (m₁ + m₂)
- v₂ = [(m₂ - m₁)u₂ + 2m₁u₁] / (m₁ + m₂)
Where m₁ and m₂ are the masses, u₁ and u₂ are the initial velocities, and v₁ and v₂ are the final velocities after collision. These formulas are essential for solving exam problems.
5. What is the main difference between 1D and 2D elastic collisions?
1D elastic collisions occur along a single straight line, while 2D elastic collisions involve motion and momentum conservation in two perpendicular directions. The main differences are:
- 1D: Only one direction (e.g., head-on collisions); only one momentum equation is used.
- 2D: Both x and y components are considered; two separate momentum equations are needed.
- Complexity: 2D problems are generally more complex and require vector analysis.
This distinction is important for JEE and competitive exam preparation.
6. If both masses are equal, what happens to their velocities after a 1D elastic collision?
When both masses are equal in a one-dimensional elastic collision, they simply exchange their velocities. That is:
- The first object takes the velocity of the second object
- The second object takes the velocity of the first
This is a classic result and can be quickly recalled for MCQs and direct numerical questions.
7. Can you provide derivation PDF/notes for 1D elastic collision?
Yes, comprehensive Class 11 Physics notes and derivation PDFs covering one-dimensional elastic collisions are available for download. These include:
- Step-by-step derivations
- Key formula sheets
- Sample solved questions
These study resources are essential for thorough JEE and Class 11 preparation.
8. What is the difference between elastic and inelastic collisions?
The main difference is that elastic collisions conserve both momentum and kinetic energy, while inelastic collisions only conserve momentum. Key points include:
- Elastic: No loss in total kinetic energy; objects bounce off each other
- Inelastic: Kinetic energy is lost as heat, sound, or deformation; objects may stick together (perfectly inelastic)
Understanding this distinction is crucial for physics exams and problem-solving.
9. What are some common mistakes students make while writing the derivation of elastic collision in one dimension?
Common mistakes in derivation of 1D elastic collision include:
- Mixing up initial and final velocities
- Forgetting to conserve both momentum and kinetic energy
- Incorrect sign conventions for direction
- Errors in algebraic manipulation
Carefully follow every step and double-check formulas during exams for full marks.
10. Which of the following collisions is one-dimensional?
A one-dimensional collision is an event where two objects move along the same straight line before and after the collision. Common examples are:
- Head-on collision between two carts on a track
- Collisions between billiard balls moving in a straight line
Recognizing the type of collision is vital for choosing the correct conservation equations in physics problems.
