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 → dynamics ↓


Motion in one dimension


Motion in one dimension, also known as linear motion, is a fundamental concept in kinematics, a branch of classical mechanics. It deals with objects moving in a straight line. This topic is important because it lays the foundation for understanding more complex motion in two or three dimensions. In one-dimensional motion, we are mainly concerned with concepts such as displacement, velocity, acceleration, and the equations of linear motion.

Basic concepts

Position and displacement

The position of an object in motion is its location at any given time. Imagine you have a number line, and you can locate an object by pointing to a number. Position is a vector quantity, which means it has both magnitude and direction. However, in one-dimensional motion, direction is usually represented by a positive or negative sign.

Object (Status = 100) New Status

Displacement is the change in the position of an object. It is also a vector quantity, which means it has both magnitude and direction. It is given by the formula:

Displacement = Final Position - Initial Position

If an object moves from a position of 100 m to a position of 300 m, the displacement will be 200 m in the positive direction.

Displacement = 200 m

Velocity

Velocity is a vector quantity that describes the rate of change of position relative to time. In simple terms, it tells us how fast an object is moving and in what direction. The formula for average velocity is:

Velocity = Displacement / Time

For example, if a car travels 200 meters in 20 seconds, the average velocity is:

Velocity = 200 m / 20 s = 10 m/s

The direction of velocity is the same as the direction of displacement. If you are moving in the negative direction, your velocity will also be negative.

Acceleration

Acceleration is a vector quantity that represents the rate of change of velocity with respect to time. It tells us how quickly something is speeding up or slowing down. The formula for average acceleration is:

Acceleration = Change in Velocity / Time

Consider a car that increases its velocity from 0 m/s to 20 m/s in 10 s. The average acceleration is:

Acceleration = (20 m/s - 0 m/s) / 10 s = 2 m/s²

Equations of motion

In one-dimensional motion, there are three main equations of motion that describe the relationship between displacement, velocity, time, and acceleration. These equations are derived based on the assumption of constant acceleration, which simplifies many analyses.

First equation of motion

The first equation of motion gives the relation between initial velocity, final velocity, acceleration and time. It can be expressed as:

v = u + at

Here, v is the final velocity, u is the initial velocity, a is the acceleration, and t is time.

Second equation of motion

This equation connects displacement with initial velocity, time, and acceleration:

s = ut + (1/2)at²

In this equation, s represents displacement.

Third equation of motion

The third equation of motion relates the square of the final velocity to the square of the initial velocity, acceleration, and displacement:

v² = u² + 2as

Text example

Example 1: A car trip

Imagine a car moving in a straight line. The car starts from a stationary state, reaches a speed of 20 m/s, and then stops. Let us analyze the motion at different stages using the equations and concepts we have learned.

Step 1: Acceleration

  • Initial velocity, u = 0 m/s
  • Final velocity, v = 20 m/s
  • Acceleration, a = 2 m/s²
  • Time taken, t = ?

We can solve for t using the first equation of motion v = u + at.

20 = 0 + 2t
t = 10 s

Step 2: Uniform motion

  • Velocity, v = 20 m/s
  • Time, t = 5 s

During this phase, the car travels at a constant velocity. Displacement can be calculated as:

s = v * t = 20 m/s * 5 s = 100 m

Phase 3: Slowdown

  • Initial velocity, u = 20 m/s
  • Final velocity, v = 0 m/s
  • Acceleration (deceleration in this case), a = -4 m/s²

Using the first equation, v = u + at, solve for t.

0 = 20 + (-4)t
t = 5 s

Example 2: Dropping the ball

Suppose a ball is dropped from a height with an initial velocity of 0. The acceleration due to gravity is about 9.8 m/s² in the negative direction.

  • Initial velocity, u = 0 m/s
  • Acceleration, a = 9.8 m/s²
  • Displacement, s = -45 m (downwards)

Using the third equation of motion, v² = u² + 2as, we can find the final velocity, v.

v² = 0 + 2 * 9.8 * 45
v² = 882
v = √882 ≈ 29.7 m/s

Visualization of one-dimensional motion

A common way to represent motion is to plot the motion over time on a graph. Consider a simple thought experiment in which an object moves along a straight path.

Example: Uniform velocity

Imagine an object moving at a constant velocity of 10 m/s for 10 seconds. When this is plotted on a position-time graph, a straight line is obtained.

Post Time t = 10s, position = 100m

This graph implies that with every passing second, the position of the object moves by 10 meters, which shows constant velocity.

Observations

  • If the graph is a straight line, it represents uniform motion.
  • The slope of the line gives the velocity of the object.

Conclusion

Understanding motion in one dimension is fundamental because it allows us to understand the basic principles of dynamics. Key concepts include displacement, velocity, and acceleration, and their relationships are described through the equations of motion. These ideas are not only theoretical, but also have practical applications in everyday tasks such as driving a car, playing sports, or even walking to a destination.

The simplicity of one-dimensional motion allows us to establish a baseline understanding before moving on to more complex kinematic problems involving two and three dimensions. This foundation is important for anyone who wants to explore or engage in physics and engineering.


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Motion in one dimension

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