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


In the study of classical mechanics, two fundamental concepts are momentum and collision. These concepts are crucial to understanding how objects in our world behave and interact, helping us predict the outcome of countless scenarios, from car accidents to the motion of celestial bodies. Let's take a deeper look at these topics, starting with a simple explanation of momentum.

Understanding momentum

Momentum is a measure of the motion of an object and is defined as the product of an object's mass and velocity. This relationship can be expressed in the formula:

p = mv

Where:

  • p is the momentum,
  • m is the mass of the object, and
  • v is the velocity of the object.

Momentum is a vector quantity, which means it has both magnitude and direction. This is important because an object moving east at a certain speed has different momentum than an object moving west at the same speed.

Example of calculating momentum

Suppose a car of mass 1200 kg is moving east at a speed of 20 m/s. The speed can be calculated as follows:

p = mv = 1200 kg * 20 m/s = 24000 kg*m/s to the east

This example shows how important the direction of velocity is in describing the direction of momentum.

Conservation of momentum

One of the major principles related to momentum is the law of conservation of momentum. This law states that in a closed system (one that is not affected by external forces), the total momentum remains constant. It is mathematically written as:

∑p initial = ∑p final

This principle is especially important when analyzing collisions and interactions between objects.

Types of collisions

Collisions are events where two or more objects exert forces on each other for a relatively short period of time. Collisions can be classified into different types based on the conservation of kinetic energy. There are two main types of collisions:

  1. Elastic collision
  2. Inelastic collision

Elastic collision

In an elastic collision both momentum and kinetic energy are conserved. This means that the total kinetic energy remains the same before and after the collision. Such collisions are characteristic of atoms and subatomic particles. Consider the following example of an elastic collision:

Imagine two billiard balls colliding in a frictionless environment. Before the collision:

Ball 1: mass = 1 kg, velocity = 2 m/s 
Ball 2: mass = 1 kg, velocity = -2 m/s

After the collision, they exchange velocities:

Ball 1: velocity = -2 m/s (after) 
Ball 2: velocity = 2 m/s (after)

In this example, despite the change in direction, the kinetic energy and momentum remain constant before and after the collision.

Inelastic collision

In an inelastic collision, momentum is conserved, but kinetic energy is not. A portion of the energy is converted into other forms of energy such as sound, heat, or deformation energy. A perfectly inelastic collision is a situation in which the colliding objects stick together after impact, and move as a single unit.

Consider two clay balls that collide and stick together:

Ball A: mass = 1 kg, velocity = 3 m/s 
Ball B: mass = 1 kg, velocity = 0 m/s

After the collision, they stick together to form a 2 kg mass:

After collision, combined mass velocity: 
v = (m A v A + m B v B ) / (m A + m B )
v = (1 kg * 3 m/s + 1 kg * 0 m/s) / (1 kg + 1 kg)
v = 1.5 m/s

Impulse and force in collision

Impulse is a concept closely related to momentum and is defined as the change in momentum of an object when a force is applied over a time interval. Impulse is calculated as follows:

Impulse = Δp = F * Δt

Where F is the force and Δt is the duration of application of the force. Impulse can be visualized by considering that padding in a car's dashboard increases the time of application of the force, which reduces the force, thus reducing the chance of injury.

Consider a scenario where a basketball is tossed. Let's say a force is applied for 0.1 seconds that causes a change in speed from 2 kg*m/s to -2 kg*m/s:

Impulse = Δp = p final - p initial 
= -2 kg*m/s - 2 kg*m/s 
= -4 kg*m/s

Impulse measures the effectiveness of a force in changing the momentum of an object.

Illustration of conservation of momentum

To make conservation of momentum more clear, consider the following simplified example:

month 1 month 2

In this illustration, two balls are moving toward each other. When they collide, their momentum changes to reflect conservation principles, with no external forces.

Applications of momentum and collision

The principles of motion and collision are used in a variety of fields. For example, in vehicle safety engineering, understanding the dynamics of inelastic collisions helps design crumple zones that absorb crash energy, thereby protecting passengers.

In sports, players apply these principles intuitively. Consider pool players who accurately predict the direction of a ball after a collision or boxers who use evasive techniques to alter the impulse and protect themselves.

Conclusion

The study of momentum and collision is important in classical mechanics and helps simplify complex interactions between objects by focusing on conservation laws. Mastering these concepts gives us the power to analyze, predict, and influence physical scenarios in everyday life, ensuring progress from technology to security systems.


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Speed and collisions

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