• Newton's Third Law of Motion

    Newton's third law states: To every action there is an equal and opposite reaction. Newton's third law implies that forces always come in pairs. In fact, these force pairs have the same origin and nature. If one is gravitational, then so is the other;...

  • Newton's Second Law of Motion

    Newtons first law of motion states that a body subject to no net external forces continues in a state of uniform motion. His second law of motion in a sense makes his first law complete by telling us what will happen in the event of a body being...

  • Newton's Three Laws of Motion

    Newtons three laws of motion govern the basis of how things move. At the basis of the three laws is the concept of force. Newton's First Law of Motion A rigid body subject to no net external force continues in a state of uniform motion. No net force so...

  • Proof of Newton's First Law of Motion

    Theorem A body continues in a state of uniform motion unless acted on by an external force. Proof Let a body consist of particles of mass at position vectors The equation of motion for particle is The force acting on the body is The position vector of...

  • Mechanical Determinism

    Newtonian mechanics is a complete system. Given a state of a system of particles and the mass, velocity and position of each particle, the state of the system – the position and velocity of each particle can be determined at any subsequent or previous...

  • University Maths Notes - Number Theory

  • The Navier - Stokes Equation

    The Navier – Stokes equation models the behaviour of a fluid element because of the forces acting on it, including viscous forces. When the viscous forces are ignored, the equations become Euler's equation. The equation is difficult to solve, and...

  • The Molecular Model of an Ideal Gas

    An ideal gas is a model of a gas that treats the gas molecules as points of infinitesimally small size that obey all Newton's Laws of motion and undergo only elastic collisions with each other and the walls of the container. In detail, the assumptions...

  • University Maths Notes - Vector Calculus

  • University Maths Notes - Topology

  • University Maths Notes - Tensors

  • Motion of a Projectile Subject to Air Resistance Proportional to Velocity

    Suppose a projectile is moving subject only to the force of gravity and air resistance proportional to speed. Applying Newton's Second Law horizontally gives \[m \frac{dv_x}{dt}= -kv_x\] Separate variables to give \[\frac{dv_x}{v_x} = - \frac{k}{m} dt\]...

  • Proof of Conservation of Angular Momentum

    Theorem Angular momentum is conserved for a body subject to no net external torque. Proof Let a body consist of particles at position vectors The motion of each particle is determined by Newton's second law or Taking the cross product with and summing...

  • Uniform Circular Motion

    The phrase 'uniform circular motion' describes planar motion in a circle at constant speed. If the motion takes place on the Earth, this almost certainly means that the motion is horizontal. For example a mass suspended at the end of a string may be...

  • Starting to Move

    Jet engines operate by producing a hot gas and expelling it from the exhaust of the engine at high speed. The exhaust gases have a large momentum and need a large force to be exerted on them to obtain this speed. Newton's Third Law of motion then tells...

  • Second Order Autonomous Systems

    Almost all Newtonian mechanical systems are second order or higher. For a second order system, two variables are needed to model the system and locate a given point in phase space. In Cartesian coordinates the equations of motion take the form and...

  • Equation if Motion For an Inviscid, Incompressible Fluid

    For an incompressible fluid, so If the body force per unit volume is applying Newton's Second Law to a cube of unit volume to obtain We can also write where is the pressure in the fluid. Equating these gives (1) Consider the motion of a particle in the...

  • Air Bags, Seat Belts and Crumple Zones

    Car crashes dont kill people - FORCES KILL PEOPLE. People die in car crashes because impact is too often instant, and the force of impact too large. If the time of impact can be prolonged, then the acceleration and the force of impact can be reduced....

  • Proving the Formula for the Kinetic Energy of a Body

    If a resultant force acts on a body initially at rest, the speed of the body will increase. The force will do work on the body. If there is no resistance to motion, all the work done by the force will be expressed in an increase of kinetic energy of...

  • Kepler's Laws of Planetary Motion

    Kepler's laws of planetary motion accurately explain and predict the motion of the planets. He made observations over many years, and was able two deduce three mathematical laws. 1. The orbit of each planet about the Sun is an ellipse with the Sun at...


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