Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Mechanics → dynamics ↓


Motion in two and three dimensions


Motion is a fundamental concept in physics. It helps us describe how objects move in space over time. While one-dimensional motion gives information about motion in a straight line, real-world scenarios often involve motion that occurs in two or three dimensions. In this lesson, we will explore motion in two and three dimensions using simple language and examples. We will touch on vectors, coordinate systems, velocity, acceleration, and projectile motion, which are essential components of motion in two and three dimensions.

Vectors and coordinate systems

Before diving into motion in two and three dimensions, it is necessary to understand vectors and coordinate systems because motion is often described using them. A vector is a quantity that has both magnitude (size) and direction. Examples of vector quantities include displacement, velocity, and acceleration.

Coordinate systems help us visualize motion. The most common coordinate system is the Cartesian coordinate system, which uses perpendicular axes, represented as x, y, and z axes. In two dimensions, motion is described using x and y axes.

        | y | / | /θ |/____ x

Motion in two dimensions

When we talk about motion in two dimensions, we usually mean motion on a plane. An example of this is a car moving on a flat road or an athlete running on a track. Movements in the horizontal and vertical directions are mapped onto x and y axes.

Two-dimensional displacement

Displacement in two dimensions is often represented as a straight line from one point to another on a plane. Consider a bird flying from point A to point B on a map, represented as coordinates. If A is at (x₁, y₁) and B is at (x₂, y₂), then the displacement vector Δr from A to B is:

        Δr = (x₂ - x₁)i + (y₂ - y₁)j

Velocity in two dimensions

Just like displacement, velocity has two components: horizontal and vertical. The velocity vector can be given as:

        v = vₓi + vᵧj

where vₓ is the velocity component in x direction and vᵧ is the velocity component in y direction.

Acceleration in two dimensions

Similarly, acceleration in two dimensions also has horizontal and vertical components. The acceleration vector is:

        a = aₓi + aᵧj

Where aₓ is the acceleration in x direction and aᵧ is the acceleration in y direction.

Projectile motion

Projectile motion is a common example of motion in two dimensions. It occurs when an object is thrown or released into the air and moves only under the influence of gravity. The path taken by a projectile is called its trajectory. For a projectile launched at an angle θ with initial velocity v₀, its motion can be analyzed as follows:

The horizontal component of velocity is v₀ cos(θ) and the vertical component is v₀ sin(θ). Horizontal motion is constant velocity motion, while vertical motion is uniformly accelerated motion due to gravity.

The horizontal position x and vertical position y at time t can be given as follows:

        x = (v₀ cos(θ)) * t
        y = (v₀ sin(θ)) * t - (1/2)gt²

Motion in three dimensions

In reality, objects often move in three-dimensional space, requiring us to describe motion using three axes: x, y, and z. Examples include airplanes flying in the sky and bees buzzing in a garden.

Three-dimensional displacement

Displacement in three dimensions involves three components, one for each axis. If an object moves from a point (x₁, y₁, z₁) to another point (x₂, y₂, z₂), its displacement vector is:

        Δr = (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k

Velocity in three dimensions

The velocity vector in three dimensions can be described as follows:

        v = vₓi + vᵧj + vzk

where vₓ, vᵧ and vz are the velocity components corresponding to x, y and z axes, respectively.

Acceleration in three dimensions

Like displacement and velocity, acceleration also has three components in three-dimensional space. The acceleration vector is given by:

        a = aₓi + aᵧj + azk

Example of motion in three dimensions

Suppose a drone is flying from point (0, 0, 0) to point (3, 4, 5). Its displacement vector is:

        Δr = (3 - 0)i + (4 - 0)j + (5 - 0)k = 3i + 4j + 5k

Conclusion

Understanding motion in two and three dimensions is crucial to describing and analyzing the motion of objects in the real world. By breaking down motion into its vector components and applying these principles, we can gain deep insights into the mechanics that govern our universe. Whether it's a car driving down a curved road or a cupcake being thrown across the room for fun, these fundamental concepts give us the tools to handle complex motion scenarios with precision.


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Motion in two and three dimensions

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