Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Speed and collisions ↓


Elastic and inelastic collision


In the field of classical mechanics, it is fundamental to understand the nature of collisions. Collisions occur in various forms and can be broadly classified into two types: elastic and inelastic collisions. These collisions are governed by the principles of momentum and energy conservation. The aim of this exposition is to explain these concepts with clarity and comprehensiveness.

Momentum and its conservation

Momentum is an important concept in physics that helps describe the motion of objects. It is defined as the product of an object's mass and velocity:

momentum (p) = mass (m) * velocity (v)

This quantity is a vector, which means it has both magnitude and direction. Conservation of momentum is a fundamental principle that states that the total momentum of a closed system remains constant if no external forces act on it.

For example, imagine two ice skaters pushing each other on a frictionless ice surface. If the system is closed and there are no external forces acting on the skaters, their total momentum remains the same before and after the push.

Elastic collision

Elastic collisions are those in which both momentum and kinetic energy are conserved. This means that the total kinetic energy of a system is the same before and after the collision. In everyday life, perfectly elastic collisions are rare, but they often occur with subatomic particles or in controlled experiments.

Visual example of an elastic collision

Suppose two billiard balls are colliding on a table. Let's imagine this:

A B

Ball A (blue) moves towards ball B (red). Before the collision, both have certain velocities. After the collision, they bounce back, maintaining the same total kinetic energy and momentum:

m A * v A,initial + m B * v B,initial = m A * v A,final + m B * v B,final
0.5 * m A * v A,initial ² + 0.5 * m B * v B,initial ² = 0.5 * m A * v A,final ² + 0.5 * m B * v B,final ²

This illustration assumes perfect elasticity, where no kinetic energy is converted to sound, heat, or other forms.

Inelastic collision

Unlike elastic collisions, kinetic energy is not conserved in inelastic collisions. However, momentum is still conserved. In an inelastic collision, some of the kinetic energy is usually converted into other energies such as heat or sound.

A perfectly inelastic collision is an extreme situation in which the maximum amount of kinetic energy is lost. After the collision the objects stick together and move with a constant velocity.

Visual example of an inelastic collision

Imagine two cars collide and stick to each other. Let's imagine this scenario:

1 2

Car 1 (green) and car 2 (orange) move towards each other. When they collide, they stick together and move forward as a unit. While momentum is conserved:

m 1 * v 1,initial + m 2 * v 2,initial = (m 1 + m 2 ) * v final

Kinetic energy is lost due to deformation or heat dissipation:

initial_k.e. > final_k.e.

This type of collision is common in real-world scenarios, such as motor vehicle accidents.

Mathematical representation and applications

Understanding the mathematical formulation of collisions equips us with the ability to predict outcomes. Let us take a deeper look at the calculations involved in elastic and inelastic collisions.

Elastic collision calculation

For an elastic collision between two objects, the conservation laws can be expressed as:

Motion:

m 1 * v 1,initial + m 2 * v 2,initial = m 1 * v 1,final + m 2 * v 2,final

Kinetic energy:

0.5 * m 1 * v 1,initial ² + 0.5 * m 2 * v 2,initial ² = 0.5 * m 1 * v 1,final ² + 0.5 * m 2 * v 2,final ²

These equations can be solved simultaneously to find the final velocities of the two objects after the collision. Algebraic manipulation usually involves substitution or the application of quadratic formulas.

Inelastic collision calculation

For an inelastic collision, momentum is conserved but kinetic energy is not. The mathematical formulation focuses on momentum conservation:

m 1 * v 1,initial + m 2 * v 2,initial = (m 1 + m 2 ) * v final

This equation gives the final velocity of the combined masses just after a direct collision.

Examples and applications in real life

To see these concepts in a real-world context, let's examine some examples and applications of elastic and inelastic collisions.

Elastic collision example: billiards

Billiards is a classic example of elastic collision, where balls collide with each other on a pool table. Billiards is not perfectly elastic, but closely resembles elastic collision, due to friction and sound.

Consider a cue ball hitting another ball directly:

m cue * v cue,initial = m cue * v cue,final + m target * v target,final

Energy and momentum conservation principles allow post-collision velocities to be predicted.

Example of an inelastic collision: automobile accidents

Vehicle collisions are often examples of inelastic collisions. These involve significant deformation, conversion of kinetic energy into heat, sound, and structural changes.

In case of a rear-end collision:

m car1 * v car1,initial + m car2 * v car2,initial = (m car1 + m car2 ) * v combined,final

The above equation estimates the final velocity after impact.

Scientific and technological applications

  • Particle Physics: Elastic collisions play an important role in particle accelerators, where subatomic particles collide and energies are studied.
  • Material Testing: Inelastic collision strength tests of materials help in analysing the deformation under stress.
  • Safety features: Understanding inelastic collision behavior guides automotive safety designs, such as energy-absorbing crumple zones.

Conclusion

Elastic and inelastic collisions are integral to the study of classical mechanics, with wide implications ranging from theoretical physics to practical engineering. By conserving momentum and analyzing energy transformations, collision dynamics reveals the fundamental interactions within closed systems.

Mastering these concepts enables scientists and engineers to innovate and refine technologies, yielding both theoretical insights and concrete advances.


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Elastic and inelastic collision

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