PHY1002 Geometrical Optics, Waves, and Mechanics

Geometrical Optics, Waves, and Mechanics §

Module Co-ordinator: TKouloukas@lincoln.ac.uk (Dr Theodoros Kouloukas, Room 3303) This module will provide an introduction to the fundamentals of waves, geometrical optics and mechanics, including their mathematical foundations. The module syllabus is designed and delivered and in such a way that no prior knowledge of physics is required.

Formulae

Course Components §

1. Coursework Assignment (10%)
2. In-Class Assignment (25%)
3. Cengage Numerical Assignment (15%)
4. Final Exams (50%)

Outline Syllabus §

• Mechanics (taught by Theodoros): - Pre-requisite(s): Calculus & Vectors.
  • Motion in one, two and three dimensions.
  • Newton’s laws.
  • Work, kinetic and potential energy.
  • Conservation laws.
  • Rotation and angular momentum.
  • Newton’s law of gravity.
  • Simple harmonic motion, pendulum.
• Waves (taught by Theodoros):
  • Simple wave motion, periodic waves, sound waves, waves in three dimensions.
  • Diffraction.
  • Doppler effect.
  • Superposition of waves.
  • Waves on strings.
  • Wave packets and group velocity.
• Geometrical Optics (taught by Andrei: A combination of mathematics and technical drawing):
  • Huygens’s construction, Fermat’s Principle (1662), Reflection and Refraction.
  • Mirrors and lenses. Interference and diffraction at single and multiple slits.
  • Fraunhofer and Fresnel diffraction, diffraction gratings.

Learning Outcomes §

• LO1 Mathematically solve simple problems of mechanics, optics and waves.
• LO2 Construct multi-step solutions of problems of mechanics, optics and waves.
• LO3 Formulate main laws of mechanics, optics and waves.

Flashcards §

Automatically generated into Anki TARGET DECK University::PHY1002 Geometrical Optics, Waves, and Mechanics

• STARTI [Basic] What is the PHY1002 module? Back: Geometrical Optics, Waves, and Mechanics. ENDI
• STARTI [Basic] Question: What is Huygens’s Principle? Back: Huygens’s Principle, also known as Huygens’s construction, is a method of analysis applied in wave theory. It proposes that each point on a wave front is a source of secondary wavelets that spread out forward at the speed of the wave. The new position of the wave front is given by the surface tangent to these secondary wavelets. ENDI
• STARTI [Basic] Question: Who proposed Huygens’s Principle? Back: Huygens’s Principle was proposed by the Dutch physicist Christiaan Huygens in the 17th century. ENDI
• STARTI [Basic] Question: How did Huygens view light? Back: Huygens proposed that light should be considered as a form of wave motion, rather than a stream of particles. This idea was contrary to Newton’s corpuscular theory of light. ENDI
• STARTI [Basic] Question: What are wave fronts in Huygens’s Construction? Back: Wave fronts are considered as sources of secondary wavelets that spread out in the forward direction at the speed of the wave. ENDI
• STARTI [Basic] Question: How is the new wave front determined in Huygens’s Construction? Back: The new position of the wave front is given by the surface that is tangent to the secondary wavelets. ENDI
• STARTI [Basic] Question: What phenomena can be explained using Huygens’s Principle? Back: Huygens’s Principle can explain various phenomena related to wave propagation, including refraction, reflection, diffraction, and interference of waves. ENDI
• STARTI [Basic] Question: How does Huygens’s Principle explain refraction? Back: The change in speed of light when it passes from one medium to another can be explained using Huygens’s Principle, leading to Snell’s Law of Refraction. ENDI
• STARTI [Basic] Question: How does Huygens’s Principle explain diffraction? Back: Diffraction, the bending of light around edges or through openings, can be explained using Huygens’s Principle. ENDI
• STARTI [Basic] Question: What is the legacy of Huygens’s Principle? Back: Huygens’s Principle laid the groundwork for the wave theory of light, which was later developed and refined by other scientists. It led to the modern understanding of light as exhibiting both wave-like and particle-like properties. ENDI
• STARTI [Basic] Question: Why is Huygens’s Principle significant in wave theory and optics? Back: Huygens’s Principle is a fundamental concept in wave theory and optics. It forms an integral part of our understanding of wave phenomena, including the behaviour of light. ENDI
• STARTI [Basic] Question: What is Fermat’s Principle of Least Time? Back: Fermat’s Principle of Least Time, also known as the law of least time, states that of all the possible paths that light could take to travel from one point to another, it takes the path which requires the least time. ENDI
• STARTI [Basic] Question: Who proposed Fermat’s Principle of Least Time? Back: Fermat’s Principle of Least Time was proposed by Pierre de Fermat in 1662. ENDI
• STARTI [Basic] Question: How does Fermat’s Principle explain the path of light in a homogeneous medium? Back: In a homogeneous medium, where the speed of light is constant, Fermat’s Principle explains that light travels in a straight line because it takes the path which requires the least time. ENDI
• STARTI [Basic] Question: How does Fermat’s Principle account for the bending of light (refraction)? Back: Fermat’s Principle accounts for the bending of light (refraction) when it passes from one medium to another with a different refractive index. As the speed of light changes between the media, it takes a path that minimizes travel time. ENDI
• STARTI [Basic] Question: How does Fermat’s Principle prove the law of reflection? Back: Using Fermat’s Principle, it can be shown that the path light takes (obeying the law of reflection) is the shortest distance between two points, thus requiring the least time. This proves the law of reflection. ENDI
• STARTI [Basic] Question: Who was Pierre de Fermat? Back: Pierre de Fermat was a lawyer by profession and a mathematician by passion who lived from 1607 to 1665. He is best known for Fermat’s Last Theorem and his principle of least time in optics. ENDI
• STARTI [Basic] Question: How is Fermat’s Principle connected to Ibn Al-Haytham? Back: An early version of the principle of least time was proposed by the Arab physicist Ibn Al-Haytham in the 11th century. His work on optics influenced later scientists, including Fermat. ENDI
• STARTI [Basic] Question: What is the significance of Fermat’s Principle of Least Time in optics? Back: Fermat’s Principle of Least Time is a key postulate in the study of light and its behaviour. It provides an explanation for why light travels the way it does, whether in straight lines or bending when it enters a new medium. ENDI
• STARTI [Basic] Question: What is the Law of Refraction also known as? Back: The Law of Refraction is also known as Snell’s Law. ENDI
• STARTI [Basic] Question: What does Snell’s Law describe? Back: Snell’s Law describes how light or any other wave changes direction when it passes from one medium to another with a different refractive index. ENDI
• STARTI [Basic] Question: State the mathematical representation of Snell’s Law. Back: Snell’s Law is represented as: (n_1\sin(\theta_1) = n_2\sin(\theta_2)), where (n_1) and (n_2) are the refractive indices of the first and second mediums, respectively, and (\theta_1) and (\theta_2) are the angles of incidence and refraction, respectively. ENDI
• STARTI [Basic] Question: How is the refractive index related to the speed of light in a medium? Back: The refractive index (n) is given by (n=\frac{c}{v}), where (v) is the velocity of light in the medium. ENDI
• STARTI [Basic] Question: Who originally discovered the Law of Refraction? Back: The Law of Refraction, or Snell’s Law, was originally discovered by the Persian scientist Ibn Sahl in the 10th century. ENDI
• STARTI [Basic] Question: How is the Law of Refraction applied in lens design? Back: The Law of Refraction is used in the design of lenses for eyeglasses, cameras, telescopes, and other optical devices. ENDI
• STARTI [Basic] Question: How is the Law of Refraction relevant to fiber optics? Back: The law of refraction is used in the design and understanding of fiber optics, which are used for high-speed data transmission. ENDI
• STARTI [Basic] Question: What principle do prisms operate on? Back: Prisms work on the principle of refraction and are used to split white light into its constituent colors. ENDI
• STARTI [Basic] Question: What are the two types of refraction? Back: The two types of refraction are Regular Refraction and Total Internal Reflection. ENDI
• STARTI [Basic] Question: What is Total Internal Reflection? Back: Total Internal Reflection occurs when light attempts to move from a denser medium to a less dense medium, and the angle of incidence is greater than the critical angle. In this case, the light is entirely reflected back into the denser medium. ENDI
• STARTI [Basic] Question: Why is understanding the Law of Refraction important? Back: Understanding the Law of Refraction is crucial for many fields, including physics, engineering, and technology design. It helps in understanding how light and other waves behave as they transition from one medium to another. ENDI
• STARTI [Basic] Question: What does the Law of Reflection describe? Back: The Law of Reflection explains how light behaves when it encounters a reflective surface, especially a mirror-like surface. ENDI
• STARTI [Basic] Question: State the main points of the Law of Reflection. Back: 1. The angle of incidence (θ1) is equal to the angle of reflection (θ2). 2. The incident ray, the reflected ray, and the normal line are all in the same plane. ENDI
• STARTI [Basic] Question: How is the Law of Reflection represented mathematically? Back: The Law of Reflection is represented as: (\theta_1=\theta_2). ENDI
• STARTI [Basic] Question: How is the Law of Reflection applied in mirror design? Back: The shape and positioning of mirrors in optical instruments like telescopes, periscopes, and cameras are based on the Law of Reflection. ENDI
• STARTI [Basic] Question: How is the Law of Reflection used in building and architectural design? Back: The law of reflection is used in designing buildings to optimize natural light. ENDI
• STARTI [Basic] Question: Why is the Law of Reflection important in radar and sonar technology? Back: The law of reflection is used in radar and sonar technology to calculate the position and distance of objects. ENDI
• STARTI [Basic] Question: What are the two types of reflection? Back: The two types of reflection are Specular (or Regular) Reflection and Diffuse Reflection. ENDI
• STARTI [Basic] Question: Describe Specular (or Regular) Reflection. Back: Specular Reflection occurs when light rays hit a smooth, shiny surface like a mirror. The rays are reflected at the same angle as they hit the surface, producing a clear, sharp reflection. ENDI
• STARTI [Basic] Question: Describe Diffuse Reflection. Back: Diffuse Reflection occurs when light rays hit a rough, matte surface. The rays are scattered in many different directions, causing the reflection to be blurry or diffused. ENDI
• STARTI [Basic] Question: To what types of waves does the Law of Reflection apply? Back: The Law of Reflection applies to light, but it also applies to other types of waves, such as sound and water waves. ENDI
• STARTI [Basic] Question: Why is understanding the Law of Reflection important? Back: Understanding the Law of Reflection is critical for anyone studying or working in fields like physics, engineering, architecture, and more. It describes the behaviour of light and other waves when they encounter reflective surfaces. ENDI
• STARTI [Basic] Question: What role have mirrors and lenses played in our understanding of light and optics? Back: Mirrors and lenses have played a vital role in shaping our understanding of light and optics, from ancient times with polished metal mirrors to modern telescopes and cameras. ENDI
• STARTI [Basic] Question: How is an image formed by a flat mirror characterized? Back: An image formed by a flat mirror is always virtual, upright, and the same size as the object. ENDI
• STARTI [Basic] Question: What is the relationship between the distance of the object and the distance of the image in a flat mirror? Back: The distance of the object from the mirror equals the distance of the image from the mirror. ENDI
• STARTI [Basic] Question: How do you distinguish between a real image and a virtual image? Back: A real image is formed when light rays converge, while a virtual image is formed when light rays seem to diverge from a point. ENDI
• STARTI [Basic] Question: How does the angle of incidence relate to the angle of reflection? Back: The angle of incidence equals the angle of reflection. ENDI
• STARTI [Basic] Question: What reversal is observed in a flat mirror? Back: A left-right reversal is observed, meaning objects appear reversed in a flat mirror. ENDI
• STARTI [Basic] Question: How is the focal length of a spherical mirror related to its radius of curvature? Back: The focal length (f) is half the radius of curvature, represented by the equation (f = \frac{R}{2}). ENDI
• STARTI [Basic] Question: What kind of image does a convex mirror consistently form? Back: A convex mirror consistently forms a virtual, upright, and diminished image. ENDI
• STARTI [Basic] Question: Which equation is used to relate the object distance, image distance, and focal length of a convex mirror? Back: The equation used is (\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}). ENDI
• STARTI [Basic] Question: What are the sign conventions for object and image distance in relation to incoming and outgoing light? Back: Object distance is positive if the object is on the same side as the incoming light. Image distance is positive if the image is on the same side as the outgoing light. ENDI
• STARTI [Basic] Question: Why are ray diagrams used? Back: Ray diagrams are a graphical method used to determine the location, size, and orientation of images. ENDI
• STARTI [Basic] Question: What happens to images formed by refraction? Back: Images formed by refraction occur due to the bending of light at an interface between two different media. ENDI
• STARTI [Basic] Question: How do converging and diverging lenses differ in the type of images they produce? Back: Converging lenses can produce both real and virtual images, while diverging lenses only create virtual images. ENDI
• STARTI [Basic] Question: What is the magnification formula for images through a thin lens? Back: The magnification formula is (M = -\frac{d_i}{d_o}). ENDI
• STARTI [Basic] Question: What is a Fresnel lens? Back: A Fresnel lens is a lightweight lens featuring a series of concentric rings, which reduces both weight and required material. ENDI
• STARTI [Basic] Question: Who employed polished metals like copper to create mirrors in ancient times? Back: The Ancient Egyptians employed polished metals, such as copper, to create mirrors. ENDI
• STARTI [Basic] Question: Who is referred to as the “father of modern optics” and what is his contribution? Back: Alhazen (Ibn al-Haytham) is regarded as the “father of modern optics” because he authored the “Book of Optics” in the 11th century and made significant strides in the realms of vision, reflection, and refraction. ENDI
• STARTI [Basic] Question: What advancements in optics were made during the Renaissance? Back: During the Renaissance, concave and convex mirrors were used artistically for unique perspectives and the invention of the telescope by figures like Galileo Galilei transformed the field of astronomy. ENDI
• STARTI [Basic] Question: What aberrations can occur in lenses? Back: Two common lens aberrations are spherical aberration, caused by the spherical shape of lenses, and chromatic aberration, which arises from the dispersion of light. ENDI
• STARTI [Basic] Question: How have mirrors and lenses influenced human history? Back: Mirrors and lenses have shaped human history by their application in various fields, from art to astronomy, reflecting their paramount importance. ENDI
• STARTI [Basic] Question: What is wave optics also termed as? Back: Physical optics. ENDI
• STARTI [Basic] Question: How does wave optics differ from geometric or ray optics? Back: Wave optics delves into the intricate wave nature of light, while geometric or ray optics simplifies light’s behaviour into straight-line paths. ENDI
• STARTI [Basic] Question: Who proposed the wave theory of light in the 17th century? Back: Christiaan Huygens. ENDI
• STARTI [Basic] Question: Which experiment in the 19th century provided compelling evidence for the wave nature of light? Back: Thomas Young’s double-slit experiment. ENDI
• STARTI [Basic] Question: What is the essence of interference? Back: The interaction of waves where they combine to form larger ripples in some places and cancel each other out in others. ENDI
• STARTI [Basic] Question: What results from constructive interference in Thomas Young’s experiment? Back: Bright fringes. ENDI
• STARTI [Basic] Question: What results from destructive interference in Thomas Young’s experiment? Back: Dark fringes. ENDI
• STARTI [Basic] Question: What are the two essential conditions for interference? Back: Coherence and Monochromaticity. ENDI
• STARTI [Basic] Question: Who made significant contributions to our understanding of diffraction? Back: Augustin-Jean Fresnel. ENDI
• STARTI [Basic] Question: What does diffraction describe? Back: The bending of light waves around obstacles or the spreading of light waves as they pass through narrow openings. ENDI
• STARTI [Basic] Question: In Young’s double-slit experiment, what causes light to spread out after passing through each slit? Back: Diffraction. ENDI
• STARTI [Basic] Question: What is the formula for path difference in Young’s double-slit experiment? Back: δ = r2 - r1 = d sin θ. ENDI
• STARTI [Basic] Question: For a bright fringe, what is the relationship between path difference, slit distance, and angle of diffraction? Back: δ = d sin θbright = mλ. ENDI
• STARTI [Basic] Question: What is a diffraction grating? Back: A setup that includes multiple slits, often hundreds or thousands, spaced closely together. ENDI
• STARTI [Basic] Question: Who first proposed the idea of diffraction in crystals? Back: Max von Laue. ENDI
• STARTI [Basic] Question: Name the two types of diffraction grating. Back: Transmission Grating and Reflection Grating. ENDI
• STARTI [Basic] Question: What happens to an electromagnetic wave when it undergoes reflection from a medium with a higher index of refraction? Back: It undergoes a phase change of 180°. ENDI
• STARTI [Basic] Question: What is the interference pattern observed when a plano-convex lens is placed on a flat glass surface called? Back: Newton’s Rings. ENDI
• STARTI [Basic] Question: Who provided a quantitative measure for resolution in optics? Back: Lord Rayleigh. ENDI
• STARTI [Basic] Question: According to Rayleigh’s criterion, when are two point sources just resolvable? Back: When the center of the diffraction pattern (central maximum) of one coincides with the first minimum of the other. ENDI
• STARTI [Basic] Question: What is the direction of oscillation of a light wave called? Back: Polarization. ENDI
• STARTI [Basic] Question: Who discovered that light becomes polarized when it reflects off certain surfaces? Back: Étienne-Louis Malus. ENDI
• STARTI [Basic] Question: What method achieves polarization through materials like Polaroid? Back: Polarization by Selective Absorption. ENDI
• STARTI [Basic] Question: What is the difference between wave optics and geometric or ray optics? Back: Wave optics delves into the intricate wave nature of light, while geometric or ray optics simplifies light’s behaviour into straight-line paths. ENDI
• STARTI [Basic] Question: Who proposed the wave theory of light in the 17th century? Back: Christiaan Huygens. ENDI
• STARTI [Basic] Question: Who provided compelling evidence for the wave nature of light in the 19th century? Back: Thomas Young. ENDI
• STARTI [Basic] Question: What phenomenon does the interaction of ripples in a pond illustrate? Back: Interference. ENDI
• STARTI [Basic] Question: What are the two primary conditions required for clear interference? Back: Coherence and Monochromaticity. ENDI
• STARTI [Basic] Question: What describes the bending of light waves around obstacles? Back: Diffraction. ENDI
• STARTI [Basic] Question: Who significantly contributed to our understanding of diffraction in the 19th century? Back: Augustin-Jean Fresnel. ENDI
• STARTI [Basic] Question: What is the path difference, denoted by δ, in Young’s double-slit experiment? Back: δ = r2 - r1 = d sin θ, where d is the distance between the slits. ENDI
• STARTI [Basic] Question: For a bright fringe in interference, what is the formula for δ? Back: δ = d sin θbright = mλ, where m is the order number. ENDI
• STARTI [Basic] Question: What does a diffraction grating consist of? Back: A diffraction grating includes multiple slits, often hundreds or thousands, spaced closely together. ENDI
• STARTI [Basic] Question: Who initially described diffraction gratings in the 17th century? Back: James Gregory. ENDI
• STARTI [Basic] Question: Who realized the potential of diffraction gratings in the 19th century? Back: Joseph von Fraunhofer. ENDI
• STARTI [Basic] Question: Who first proposed the idea of crystals acting as natural diffraction gratings? Back: Max von Laue. ENDI
• STARTI [Basic] Question: What is the difference between a transmission grating and a reflection grating? Back: A transmission grating is made by cutting parallel grooves on a glass plate, while a reflection grating is made by cutting parallel grooves on a reflective material. ENDI
• STARTI [Basic] Question: What happens to an electromagnetic wave when it reflects from a higher index medium? Back: It undergoes a phase change of 180°. ENDI
• STARTI [Basic] Question: What interference phenomenon is responsible for the vibrant colours in thin films like soap bubbles? Back: Interference in Thin Films. ENDI
• STARTI [Basic] Question: Who first observed the interference pattern known as Newton’s rings? Back: Sir Isaac Newton. ENDI
• STARTI [Basic] Question: What does resolution in optics refer to? Back: The ability of an imaging system to distinguish between closely spaced objects or features. ENDI
• STARTI [Basic] Question: According to Rayleigh’s criterion, when are two point sources just resolvable? Back: When the center of the diffraction pattern (central maximum) of one coincides with the first minimum of the other. ENDI
• STARTI [Basic] Question: What direction does every light wave oscillate in? Back: This direction of oscillation is termed polarization. ENDI
• STARTI [Basic] Question: Who discovered that light becomes polarized when it reflects off certain surfaces? Back: Étienne-Louis Malus in 1808. ENDI
• STARTI [Basic] Question: How does a material like Polaroid achieve polarization? Back: Through selective absorption, transmitting light waves oscillating in one direction and absorbing those in perpendicular directions. ENDI
• STARTI [Basic] Question: Who discovered Polaroid and its properties related to polarization? Back: E. H. Land. ENDI
• STARTI [Basic] Question: What is diffraction? Back: Diffraction refers to the phenomenon wherein waves, such as light, bend around obstacles or spread out as they pass through an opening. This wave behaviour can be observed for any type of wave, including sound, water, and electromagnetic waves such as light. ENDI
• STARTI [Basic] Question: What determines the resultant pattern formed due to diffraction? Back: The resultant pattern formed due to diffraction depends on the size of the obstacle or aperture and the wavelength of the incoming wave. ENDI
• STARTI [Basic] Question: Define Fraunhofer diffraction. Back: Fraunhofer diffraction, also known as far-field diffraction, occurs when the source of light and the screen (or the observation point) are effectively at infinite distances from the diffracting aperture or obstacle. ENDI
• STARTI [Basic] Question: How is the resultant diffraction pattern in Fraunhofer diffraction described? Back: It is typically observed using parallel light beams, and the resultant diffraction pattern can be described using simple mathematical formulas. The pattern can be observed using a lens to focus the diffracted light. ENDI
• STARTI [Basic] Question: What is the formula for primary maxima and minima in single-slit Fraunhofer diffraction? Back: ( \sin \theta = \frac{m \lambda}{a} ) where (\theta) is the angle of diffraction, (\lambda) is the wavelength of light, and (m) is an integer representing the order of the fringe. ENDI
• STARTI [Basic] Question: Define Fresnel diffraction. Back: Fresnel diffraction, or near-field diffraction, takes place when either the source of light or the screen (or both) are at a finite distance from the diffracting aperture. ENDI
• STARTI [Basic] Question: What differentiates the analysis of Fresnel diffraction from Fraunhofer diffraction? Back: The analysis of Fresnel diffraction is more complicated than Fraunhofer diffraction due to the curvature and divergence of the incident wavefront. Advanced mathematical techniques, such as the Cornu spiral or the Fresnel integrals, are often employed to describe this phenomenon. ENDI
• STARTI [Basic] Question: How do the patterns produced in Fresnel diffraction vary compared to those in Fraunhofer diffraction? Back: The patterns produced in Fresnel diffraction vary in shape and size depending on the distance from the aperture, unlike the fixed pattern in Fraunhofer diffraction. ENDI
• STARTI [Basic] Question: What is a diffraction grating? Back: A diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams that travel in different directions. This splitting is based on the wavelength of light, thus allowing spectral analysis. ENDI
• STARTI [Basic] Question: What is the grating equation for the angle to the mth order maximum in diffraction grating? Back: ( d \sin \theta = m \lambda ) where (m) is an integer representing the order of the fringe. ENDI
• STARTI [Basic] Question: What is the main application of diffraction gratings? Back: Diffraction gratings are pivotal in spectroscopy for analyzing the spectral content of light. ENDI
• STARTI [Basic] Question: Who is Augustin-Jean Fresnel and what is his contribution to the field of light? Back: Augustin-Jean Fresnel was a pioneering French physicist who played a key role in the establishment of the wave theory of light. He introduced the concept of wavefronts and developed the theory of Fresnel diffraction. ENDI
• STARTI [Basic] Question: What is Joseph von Fraunhofer known for? Back: Joseph von Fraunhofer was a German physicist and optical lens manufacturer who made significant advancements in the field of spectroscopy. He observed and documented the dark lines in the sun’s spectrum, which are now known as Fraunhofer lines. His works laid the groundwork for Fraunhofer diffraction. ENDI
• STARTI [Basic] Question: Who is credited with some of the earliest works on diffraction gratings? Back: Sir David Brewster is credited with some of the earliest works on diffraction gratings. ENDI
• STARTI [Basic] Question: What is displacement, denoted by (s)? Back: Displacement is the change in position of a particle. In one dimension, it’s represented as a scalar with sign indicating direction: (s = s_f - s_i). ENDI
• STARTI [Basic] Question: Define average velocity. Back: Average velocity, (v_{avg}), is the rate of change of displacement given by (v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s_f - s_i}{t_f - t_i}). ENDI
• STARTI [Basic] Question: How is instantaneous velocity represented in terms of calculus? Back: Instantaneous velocity, (v(t)), is given by (v(t) = \frac{ds}{dt}). ENDI
• STARTI [Basic] Question: Define acceleration, (a). Back: Acceleration is the rate of change of velocity, represented as (a = \frac{dv}{dt}). ENDI
• STARTI [Basic] Question: What does the dot notation in calculus represent? Back: The dot notation represents differentiation with respect to time. For example, (\dot{s}) is velocity and (\dot{v}) or (\ddot{s}) represents acceleration. ENDI
• STARTI [Basic] Question: What is a primitive function or antiderivative? Back: A primitive function of a function (f) is a function (F) such that (F’(x) = f(x)). In mechanics, it implies finding displacement from velocity or velocity from acceleration by integrating. ENDI
• STARTI [Basic] Question: State the SUVAT equation relating displacement, initial velocity, time, and acceleration. Back: (s = ut + \frac{1}{2} a t^2). ENDI
• STARTI [Basic] Question: How can the velocity of an object under constant acceleration be calculated over time? Back: (v = u + at). ENDI
• STARTI [Basic] Question: Provide the SUVAT equation that links final velocity, initial velocity, acceleration, and displacement. Back: (v^2 = u^2 + 2as). ENDI
• STARTI [Basic] Question: How can displacement be derived using the average of initial and final velocities? Back: (s = \frac{u + v}{2} t). ENDI
• STARTI [Basic] Question: Give the equation that determines displacement considering final velocity and time under constant acceleration. Back: (s = vt - \frac{1}{2} a t^2). ENDI
• STARTI [Basic] Question: Describe the displacement vector in 3D. Back: The displacement vector is (\mathbf{s} = \langle s_x, s_y, s_z \rangle). ENDI
• STARTI [Basic] Question: What is the velocity vector in 3D and its components? Back: The velocity vector is (\mathbf{v} = \langle v_x, v_y, v_z \rangle). Components are given by (v_x = \frac{ds_x}{dt}), (v_y = \frac{ds_y}{dt}), (v_z = \frac{ds_z}{dt}). ENDI
• STARTI [Basic] Question: Define the acceleration vector in 3D and its components. Back: The acceleration vector is (\mathbf{a} = \langle a_x, a_y, a_z \rangle). Components are (a_x = \frac{dv_x}{dt}), (a_y = \frac{dv_y}{dt}), (a_z = \frac{dv_z}{dt}). ENDI
• STARTI [Basic] Question: Can SUVAT equations be applied to motion in 2D or 3D? Back: Yes, in 2D or 3D motion with constant acceleration, the SUVAT equations apply to each dimension independently. ENDI
• STARTI [Basic] Question: What is essential to remember about vectors? Back: Vectors have both magnitude and direction. In multiple dimensions, consider each component of the vector independently. ENDI
• STARTI [Basic] Question: What did Aristotle believe about motion? Back: Aristotle believed that heavenly objects moved in perfect circles with different physics than earthly objects. He also thought heavier objects fell faster and that continuous force was needed for motion. ENDI
• STARTI [Basic] Question: Who is considered the “father of modern physics” and why? Back: Galileo Galilei is often called the “father of modern physics” because of his experiments with inclined planes and his challenge to Aristotle’s beliefs about motion. ENDI
• STARTI [Basic] Question: What are Kepler’s contributions to the understanding of planetary motion? Back: Kepler described planetary motion with three laws, showing planets move in elliptical orbits and sweep out equal areas in equal times. ENDI
• STARTI [Basic] Question: Summarize Newton’s contributions to motion. Back: Newton introduced three laws of motion, the concept of force, and the law of universal gravitation. His equation (F = ma) linked force, mass, and acceleration. ENDI
• STARTI [Basic] Question: How did Einstein’s theory of relativity affect our understanding of motion? Back: Einstein’s relativity provided a framework for motion at very high speeds or in strong gravitational fields, establishing the relationship between energy and mass with (E = mc^2). ENDI
• STARTI [Basic] Question: What does quantum mechanics explain about motion? Back: Quantum mechanics explains motion at atomic and subatomic levels, introducing concepts like wave-particle duality and the uncertainty principle. ENDI
• STARTI [Basic] Question: What is displacement, represented by $s$? Back: Displacement is the change in position of a particle. In one dimension, it’s typically a scalar with the equation ( s = s_f - s_i ) where ( s_f ) is the final position and ( s_i ) is the initial position. ENDI
• STARTI [Basic] Question: Define average velocity and instantaneous velocity. Back: Average velocity is ( v_{avg} = \frac{\Delta s}{\Delta t} ) and Instantaneous velocity is ( v(t) = \frac{ds}{dt} ). ENDI
• STARTI [Basic] Question: What does acceleration, represented by $a$, denote? Back: It’s the rate of change of velocity given by ( a = \frac{dv}{dt} ). ENDI
• STARTI [Basic] Question: What does the dot notation in calculus represent? Back: It represents differentiation with respect to time. For example, ( \dot{s} ) represents velocity and ( \dot{v} ) or ( \ddot{s} ) represents acceleration. ENDI
• STARTI [Basic] Question: What is a primitive function or antiderivative? Back: A primitive function of a function ( f ) is a function ( F ) such that ( F’(x) = f(x) ). In mechanics, it means finding displacement from velocity or velocity from acceleration by integrating. ENDI
• STARTI [Basic] Question: State the SUVAT equation linking displacement, initial velocity, time, and acceleration. Back: ( s = ut + \frac{1}{2} a t^2 ). ENDI
• STARTI [Basic] Question: What is the SUVAT equation linking final velocity, initial velocity, and acceleration? Back: ( v = u + at ). ENDI
• STARTI [Basic] Question: Which SUVAT equation is derived by multiplying the second equation by ( t ) and adding to the first? Back: ( v^2 = u^2 + 2as ). ENDI
• STARTI [Basic] Question: State the SUVAT equation derived from the average of initial and final velocities multiplied by time. Back: ( s = \frac{u + v}{2} t ). ENDI
• STARTI [Basic] Question: How is displacement represented in 2D or 3D? Back: Displacement Vector is ( \mathbf{s} = \langle s_x, s_y, s_z \rangle ). ENDI
• STARTI [Basic] Question: Describe the velocity vector in 2D or 3D. Back: Velocity Vector is ( \mathbf{v} = \langle v_x, v_y, v_z \rangle ) with each component given by ( v_x = \frac{ds_x}{dt} ) and so on. ENDI
• STARTI [Basic] Question: Define the acceleration vector in 2D or 3D. Back: Acceleration Vector is ( \mathbf{a} = \langle a_x, a_y, a_z \rangle ) with each component derived similarly to velocity. ENDI
• STARTI [Basic] Question: Do the SUVAT equations apply in 2D or 3D motion with constant acceleration? Back: Yes, they apply to each dimension independently. ENDI
• STARTI [Basic] Question: What did Aristotle believe about motion? Back: Aristotle believed objects in heavens moved in perfect circles and had different physics than objects on Earth. He also thought heavier objects fell faster. ENDI
• STARTI [Basic] Question: What were Galileo Galilei’s contributions to understanding motion? Back: He contradicted Aristotle by showing objects fall at the same rate irrespective of weight (without air resistance) and laid groundwork for uniformly accelerated motion. ENDI
• STARTI [Basic] Question: What are Sir Isaac Newton’s contributions to motion? Back: Newton formulated the three laws of motion and introduced the concept of force. His equation, ( F = ma ), linked force, mass, and acceleration. ENDI
• STARTI [Basic] Question: How did Einstein’s theory of relativity contribute to our understanding of motion? Back: Einstein provided a new framework for understanding motion at very high speeds or strong gravitational fields. His equation, ( E = mc^2 ), linked energy and mass. ENDI
• STARTI [Basic] Question: Describe the difference between instantaneous velocity and average velocity using a real-world example. Back: Instantaneous velocity is the speed at a specific instant, while average velocity is the total distance covered over a period divided by the time taken. For example, a car’s speedometer shows its instantaneous velocity, but its average velocity would consider its entire journey’s time and distance. ENDI
• STARTI [Basic] Question: Regarding a ball thrown vertically upwards and then coming down, at what point of its motion does it have maximum potential energy? And maximum kinetic energy? Back: It has maximum potential energy at its highest point. It has maximum kinetic energy just before hitting the ground or just after being released. ENDI
• STARTI [Basic] Question: Define displacement in one dimension. Back: Displacement is the change in position of a particle. In one dimension, it’s typically represented as a scalar with a sign indicating direction given by ( s = s_f - s_i ), where ( s_f ) is the final position and ( s_i ) is the initial position. ENDI
• STARTI [Basic] Question: What is the formula for average velocity? Back: Average velocity is given by ( v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s_f - s_i}{t_f - t_i} ). ENDI
• STARTI [Basic] Question: Define instantaneous velocity. Back: Instantaneous velocity is given by ( v(t) = \frac{ds}{dt} ). ENDI
• STARTI [Basic] Question: What is acceleration? Back: Acceleration is the rate of change of velocity, given by ( a = \frac{dv}{dt} ). ENDI
• STARTI [Basic] Question: Describe the dot notation in calculus. Back: The dot notation represents differentiation with respect to time. For example, ( \dot{s} ) is the first derivative of ( s ) with respect to ( t ), representing velocity, while ( \dot{v} ) or ( \ddot{s} ) represents acceleration, the second derivative of ( s ) with respect to ( t ). ENDI
• STARTI [Basic] Question: What is a primitive function? Back: A primitive function (or antiderivative) of a function ( f ) is a function ( F ) such that ( F’(x) = f(x) ). In mechanics, this often involves finding displacement when given velocity or finding velocity when given acceleration by integrating. ENDI
• STARTI [Basic] Question: Provide the SUVAT equation for displacement based on initial velocity, acceleration, and time. Back: ( s = ut + \frac{1}{2} a t^2 ). ENDI
• STARTI [Basic] Question: Provide the SUVAT equation relating final velocity, initial velocity, acceleration, and displacement. Back: ( v^2 = u^2 + 2as ). ENDI
• STARTI [Basic] Question: Define the displacement vector in 2D or 3D motion. Back: Displacement Vector is ( \mathbf{s} = \langle s_x, s_y, s_z \rangle ). ENDI
• STARTI [Basic] Question: How is the velocity vector defined in 2D or 3D motion? Back: Velocity Vector is ( \mathbf{v} = \langle v_x, v_y, v_z \rangle ), where each component is defined as ( v_x = \frac{ds_x}{dt}, v_y = \frac{ds_y}{dt}, v_z = \frac{ds_z}{dt} ). ENDI
• STARTI [Basic] Question: How do SUVAT equations apply in 2D or 3D motion? Back: In 2D or 3D motion with constant acceleration, the SUVAT equations apply to each dimension independently. ENDI
• STARTI [Basic] Question: What did Aristotle believe about motion? Back: Aristotle believed that objects in the heavens moved in perfect circles and had different physics than objects on Earth. He also thought heavier objects fell faster than lighter ones and that motion required continuous application of force. ENDI
• STARTI [Basic] Question: What were Galileo Galilei’s contributions to understanding motion? Back: Galileo contradicted Aristotle by showing that objects fall at the same rate regardless of their weight (in the absence of air resistance) and laid the groundwork for understanding uniformly accelerated motion. ENDI
• STARTI [Basic] Question: Describe Sir Isaac Newton’s contributions to motion. Back: Newton formulated three laws of motion that became the foundation of classical mechanics. He introduced the concept of force and showed that the motion of objects on Earth and in space can be described by the same set of laws. His equation, ( F = ma ), linked force, mass, and acceleration. ENDI
• STARTI [Basic] Question: How did Albert Einstein reshape our understanding of motion? Back: Einstein’s theory of relativity provided a framework for understanding motion at very high speeds or in strong gravitational fields. His equation, ( E = mc^2 ), established the relationship between energy and mass. ENDI
• STARTI [Basic] Question: Explain the significance of quantum mechanics in understanding motion. Back: Quantum mechanics, developed by scientists like Max Planck, Niels Bohr, and Werner Heisenberg, explains motion at the atomic and subatomic levels, introducing concepts like wave-particle duality and the uncertainty principle. ENDI
• STARTI [Basic] Question: Calculate the acceleration of a car that accelerates uniformly from rest to 60 m/s over 10 seconds. Back: Using the formula ( a = \frac{\Delta v}{\Delta t} ), the acceleration is ( a = \frac{60 \text{ m/s}}{10 \text{ seconds}} = 6 \text{ m/s}^2 ). ENDI
• STARTI [Basic] Question: How far does an object fall from rest under gravity in 4 seconds, ignoring air resistance? Back: Using the formula ( s = \frac{1}{2} g t^2 ), where ( g ) is approximately ( 9.81 \text{ m/s}^2 ), the distance is ( s = \frac{1}{2} \times 9.81 \times 4^2 = 78.48 \text{ meters} ). ENDI
• STARTI [Basic] Question: Calculate the maximum height of a projectile launched at 30° with an initial speed of 50 m/s. Back: Using the formula ( h = \frac{u^2 \sin^2(\theta)}{2g} ), where ( u ) is the initial speed, ( \theta ) is the launch angle, and ( g ) is gravity, the maximum height is determined by this formula. ENDI
• STARTI [Basic] Question: Find the resultant velocity of a plane heading due east at 150 km/h with a wind blowing from the north at 50 km/h. Back: Using vector addition, the resultant velocity can be found using the Pythagorean theorem. ENDI
• STARTI [Basic] Question: Find the position of a drone after 5 seconds that starts at point A(2,3,4) and moves with a velocity of ( \mathbf{v} = \langle 3, -2, 1 \rangle \text{ m/s} ). Back: The new position is determined by ( B(x,y,z) = A + 5\mathbf{v} ). ENDI
• STARTI [Basic] Question: Find the particle’s acceleration at t = 2 seconds given by ( \mathbf{v}(t) = \langle t^2, 2t, 3t^3 \rangle \text{ m/s} ). Back: Acceleration is the derivative of the velocity function, ( \mathbf{a}(t) = \frac{d\mathbf{v}}{dt} ). Compute the derivative and evaluate at ( t = 2 ) seconds. ENDI
• STARTI [Basic] Question: What does Newton’s First Law state? Back: Newton’s First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. This property is known as inertia. ENDI
• STARTI [Basic] Question: How can Newton’s Second Law be expressed mathematically? Back: Newton’s Second Law can be expressed as (\vec{F} = m\vec{a}), where (\vec{F}) is the force applied, (m) is the mass of the object, and (\vec{a}) is the acceleration. ENDI
• STARTI [Basic] Question: What does Newton’s Third Law state? Back: Newton’s Third Law states that for every action, there is an equal and opposite reaction. Mathematically, this can be expressed as (\vec{F}{12} = -\vec{F}{21}), where (\vec{F}{12}) is the force applied by object 1 on object 2, and (\vec{F}{21}) is the force applied by object 2 on object 1. ENDI
• STARTI [Basic] Question: What is an orthonormal frame? Back: An orthonormal frame is a set of vectors in a vector space that are mutually orthogonal (perpendicular to each other) and all of unit length, forming a basis for the space. ENDI
• STARTI [Basic] Question: What are the conditions for a set of vectors to be orthonormal? Back: The conditions for a set of vectors to be orthonormal are orthogonality ((\vec{e}_i \cdot \vec{e}_j = 0) for (i \neq j)) and normality ((|\vec{e}_i| = 1) for all (i)). ENDI
• STARTI [Basic] Question: How do you express a vector in an orthonormal frame? Back: A vector (\vec{v}) can be expressed in an orthonormal frame as (\vec{v} = v^1\vec{e}_1 + v^2\vec{e}_2 + … + v^n\vec{e}_n), where (v^i) are the components of (\vec{v}) in this basis. ENDI
• STARTI [Basic] Question: What simplifies about a matrix representing a transformation in an orthonormal frame? Back: The matrix representing a transformation in an orthonormal frame will be orthogonal, meaning its inverse is simply its transpose, which simplifies computations. ENDI
• STARTI [Basic] Question: How do you find the components of a vector in an orthonormal basis? Back: The components (v^i) of a vector (\vec{v}) in an orthonormal basis are found by projecting (\vec{v}) onto the basis vectors: (v^i = \vec{v} \cdot \vec{e}_i). ENDI

Weeks Content §

Week 1 §

Wk1_optics.pdf Geometrical Optics W1.pdf Optics Problems Week1.pdf

Homework Optics 1 Answers to Optics Practical 1

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Week 2 §

Wk2_optics.pdf Geometrical Optics W2.pdf Optics Problems Week2.pdf

Homework Optics 2 Answers to Optics Practical 2 Optics Coursework 1

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Week 3 §

Wk3_optics.pdf Geometrical Optics W3.pdf

Homework Optics 3 Answers to Optics Practical 3 Rev_problems.pdf

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Week 4 §

Week4_Lect.pdf Practical1.pdf

Answers to Mechanics Practical 1

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Week 5 §

Week5_Lect.pdf Practical2.pdf

Answers to Mechanics Practical 2

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Week 6 §

Practical34bis.pdf

Answers to Mechanics Practical 3

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Week 8 §

Week8_Lect.pdf Practical4-5.pdf

Answers to Mechanics Practical 4

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