Topics
Units and Measurements
- Quantitative Science
- System of Units
- Derived Quantities and Units
- Rules and Conventions for Writing SI Units and Their Symbols
- Measurement of Length
- Measurement of Mass
- Measurement of Time
- Dimensions and Dimensional Analysis
- Accuracy, Precision and Uncertainty in Measurement
- Errors in Measurements>Systematic Errors
- Errors in Measurements>Random Errors
- Estimation of Errors
- Combination of Errors
- Significant Figures
- Definitions of SI Units and Constants
Mathematical Methods
- Vector Analysis
- Scalar
- Vector
- Vector Operations>Multiplication of a Vector by a Scalar
- Vector Operations>Addition and Subtraction of Vectors
- Vector Operations>Triangle Law for Vector Addition
- Vector Operations>Law of parallelogram of vectors
- Resolution of Vectors
- Multiplication of Vectors
- Scalar Product(Dot Product)
- Vector Product (Cross Product)
- Concept of Calculus
- Differential Calculus
- Integral Calculus
Motion in a Plane
- Concept of Motion
- Rectilinear Motion
- Displacement
- Path Length
- Average Velocity
- Average Speed
- Instantaneous Velocity
- Instantaneous Speed
- Acceleration in Linear Motion
- Relative Velocity
- Motion in Two Dimensions-Motion in a Plane
- Average and Instantaneous Velocities
- Acceleration in a Plane
- Equations of Motion in a Plane with Constant Acceleration
- Relative Velocity in Two Dimensions
- Projectile Motion
- Uniform Circular Motion (UCM)
- Key Parameters of Circular Motion
- Centripetal Acceleration
- Conical Pendulum
Laws of Motion
- Fundamental Principles of Motion and Mechanics
- Types of Motion
- Aristotle’s Fallacy
- Newton’s Laws of Motion
- Newton's First Law of Motion
- Newton’s Second Law of Motion
- Newton's Third Law of Motion
- Inertial and Non-inertial Frames of Reference
- Types of Forces>Fundamental Forces in Nature
- Types of Forces>Contact and Non-Contact Forces
- Types of Forces>Real and Pseudo Forces
- Types of Forces>Conservative and Non-Conservative Forces
- Types of Forces>Work Done by a Variable Force
- Work Energy Theorem
- Principle of Conservation of Linear Momentum
- Collisions
- Elastic and Inelastic Collisions
- Perfectly Inelastic Collision
- Coefficient of Restitution e
- Expressions for Final Velocities in Elastic Head-On Collision
- Loss of Kinetic Energy in Perfectly Inelastic Head-On Collision
- Collision in Two Dimensions
- Impulse of a Force
- Necessity of Defining Impulse
- Rotational Analogue of a Force: Moment of a Force Or Torque
- Couple and Its Torque
- Proof of Independence of the Axis of Rotation
- Mechanical Equilibrium
- States of Equilibrium
- Centre of Mass>Mathematical Understanding of Centre of Mass
- Centre of Mass>Velocity of Centre of Mass
- Centre of Mass>Acceleration of Centre of Mass
- Centre of Mass>Characteristics of Centre of Mass
- Centre of Gravity
Gravitation
- Concept of Gravitation
- Kepler’s Laws
- Law of Orbit or Kepler's First Law
- Law of Areas or Kepler's Second Law
- Law of Periods or Kepler's Third Law
- Newton's Universal Law of Gravitation
- Measurement of the Gravitational Constant (G)
- Acceleration Due to Gravity (Earth’s Gravitational Acceleration)
- Variation in the Acceleration>Variation in Gravity with Altitude
- Variation in the Acceleration>Variation in Gravity with Depth
- Variation in the Acceleration>Variation in Gravity with Latitude and Rotation of the Earth
- Variation in the Acceleration>Effect of the shape of the Earth
- Gravitational Potential Energy
- Expression for Gravitational Potential Energy
- Connection of Potential Energy Formula with mgh
- Potential and Potential Difference
- Escape Velocity
- Earth Satellites
- Projection of Satellite
- Weightlessness in a Satellite
- Time Period of Satellite
- Binding Energy of an Orbiting Satellite
Mechanical Properties of Solids
- Mechanical Properties of Solids
- Elastic Behavior of Solids
- Stress and Strain
- Types of Stress and Corresponding Strain
- Hooke’s Law
- Elastic Modulus>Young’s Modulus
- Elastic Modulus>Bulk Modulus
- Elastic Modulus>Modulus of Rigidity
- Elastic Modulus>Poisson’s Ratio
- Stress-strain Curve
- Strain Energy
- Hardness of Material
- Friction in Solids
- Origin of Friction
- Types of Friction>Static Friction
- Types of Friction>Kinetic Friction
- Types of Friction>Rolling Friction
Thermal Properties of Matter
- Thermal Properties of Matter
- Temperature and Heat
- Measurement of Temperature
- Absolute Zero and Absolute Temperature
- Ideal Gas Equation
- Thermal Expansion
- Linear Expansion
- Areal Expansion
- Volume Expansion
- Relation Between Coefficient of Expansion
- Specific Heat Capacity
- Specific Heat Capacity of Solids and Liquids
- Specific Heat Capacity of Gas
- Heat Equation
- Thermal Capacity
- Calorimetry
- Change of State
- Analysis of Observation>From Point A to B
- Analysis of Observation>From Point B to D
- Temperature Effects and Considerations
- Evaporation vs Boiling
- Boiling Point and Pressure
- Factors Affecting Cooking
- Sublimation
- Phase Diagram
- Gas and Vapour
- Latent Heat
- Heat Transfer
- Conduction
- Thermal Conductivity
- Coefficient of Thermal Conductivity
- Thermal Resistance
- Applications of Thermal conductivity
- Convection
- Application of Convection
- Free and Forced Convection
- Radiation
- Newton’s Law of Cooling
Sound
- Sound Waves
- Common Properties of All Waves
- Transverse Waves
- Longitudinal Waves
- Mathematical Expression of a Wave
- The Speed of Travelling Waves
- The Speed of Transverse Waves
- The Speed of Longitudinal Waves
- Newton's Formula for Velocity of Sound
- Laplace’s Correction
- Factors Affecting Speed of Sound
- Principle of Superposition of Waves
- Echo
- Reverberation
- Acoustics
- Qualities of Sound
- Doppler Effect
- Source Moving and Listener Stationary
- Listener Approaching a Stationary Source with Velocity
- Both Source and Listener are Moving
- Common Properties between Doppler Effect of Sound and Light
- Major Differences between Doppler Effects of Sound and Light
Optics
- Fundamental Concepts of Light
- Nature of Light
- Ray Optics Or Geometrical Optics
- Cartesian Sign Convention
- Reflection>Reflection from a Plane Surface
- Reflection>Reflection from Curved Mirrors
- Total Internal Reflection
- Refraction of Light
- Applications of Total Internal Reflection
- Refraction at a Spherical Surface and Lenses
- Thin Lenses and Their Combination
- Refraction at a Single Spherical Surface
- Lens Makers' Equation
- Dispersion of Light
- Analysis of Prism
- Thin Prisms
- Some Natural Phenomena Due to Sunlight
- Defects of Lenses
- Optical Instruments
- Simple Microscope or a Reading Glass
- Compound Microscope
- Telescope
Electrostatics
- Concept of Electrostatics
- Electric Charge
- Basic Properties of Electric Charge
- Additive Nature of Charge
- Quantization of Charge
- Conservation of Charge
- Force between Charges
- Coulomb’s Law
- Scalar Form of Coulomb’s Law
- Relative Permittivity or Dielectric Constant
- Definition of Unit Charge from the Coulomb’s Law
- Coulomb's Law in Vector Form
- Principle of Superposition
- Electric Field
- Electric Field Intensity Due to a Point-Charge
- Practical Way of Calculating Electric Field
- Electric Lines of Force
- Electric Flux
- Gauss’s Law
- Electric Dipole
- Couple Acting on an Electric Dipole in a Uniform Electric Field
- Electric Intensity at a Point Due to an Electric Dipole
- Continuous Charge Distribution
Electric Current Through Conductors
- Concept of Electric Currents in Conductors
- Electric Current
- Flow of Current Through a Conductor
- Drift Speed
- Ohm's Law
- Limitations of Ohm’s Law
- Electrical Power
- Resistors
- Rheostat
- A combination of resistors in both series and parallel
- Specific Resistance
- Variation of Resistance with Temperature
- Electromotive Force (emf)
- Cells in Series
- Cells in Parallel
- Types of Cells
Magnetism
- Concept of Magnetism
- Magnetic Lines of Force
- The Bar Magnet
- Magnetic Field due to a Bar Magnet
- Magnetic Field Due to a Bar Magnet at an Arbitrary Point
- Gauss' Law of Magnetism
- The Earth’s Magnetism
Electromagnetic Waves and Communication System
- Foundations of Electromagnetic Theory
- EM Wave
- Sources of EM Waves
- Characteristics of EM Waves
- Electromagnetic Spectrum
- Radio Waves
- Microwaves
- Infrared waves
- Visible Light
- Ultraviolet rays
- X-rays
- Gamma Rays
- Propagation of EM Waves
- Ground (surface) Wave
- Space wave
- Sky wave propagation
- Communication System
- Elements of a Communication System
- Commonly Used Terms in Electronic Communication System
- Modulation
Semiconductors
- Concept of Semiconductors
- Electrical Conduction in Solids
- Band Theory of Solids
- Intrinsic Semiconductor
- Extrinsic Semiconductor
- n-type semiconductor
- p-type semiconductor
- Charge neutrality of extrinsic semiconductors
- p-n Junction
- A p-n Junction Diode
- Basics of Semiconductor Devices
- Applications of Semiconductors and P-n Junction Diode
- Thermistor
- Introduction
- Definition: Relative Velocity
- Formula: Relative Velocity
- Sign Convention
- Example
- Real-Life Examples
Maharashtra State Board: Class 11
Introduction
Imagine you're walking inside a train:
- To you: You walk at 5 km/h
- To a friend outside: You move at 45 km/h (train's 40 km/h + your 5 km/h walking)
- To other passengers: You walk at 5 km/h
The same motion appears different depending on who measures it. This depends on the reference frame (viewpoint) used.
Key Idea: Motion is relative. What matters is which frame you measure from.
Maharashtra State Board: Class 11
Definition: Relative Velocity
Relative velocity is the velocity of one object as measured from another moving object's perspective.
Let:
- vA = velocity of object A (relative to ground/Earth)
- vB = velocity of object B (relative to ground/Earth)
- vAB = velocity of A relative to B (what B observes about A's motion)
Maharashtra State Board: Class 11
Formula: Relative Velocity
vAB = vA - vB
vBA = vB - vA = -vAB
Key relationship: vAB = -vBA
Maharashtra State Board: Class 11
Sign Convention
| Situation | Velocities vA, vB | Relative Velocity vAB | Interpretation |
|---|---|---|---|
| Same direction, A faster | vA > vB | Positive | A moves ahead of B; A is "overtaking" B |
| Same direction, A slower | vA < vB | Negative | A falls back behind B; B is pulling away from A |
| Opposite directions | Opposite signs | Larger magnitude | They approach each other; "closing speed" is large |
Maharashtra State Board: Class 11
Example
Example 1: Cars Moving in the Same Direction
Scenario:
- Car A moves east at 60 km/h
- Car B moves east at 40 km/h
Find: Relative velocity of A with respect to B (vAB)
Solution:
vAB = vA − vB = 60 − 40 = +20 km/h
Interpretation: Car A appears to move 20 km/h to the east relative to B. From B's perspective, A is pulling ahead at 20 km/h.
Check:
vBA = vB − vA = 40 − 60 = −20 km/h
From A's perspective, B appears to move 20 km/h backward (west).
Example 2: Objects Moving in Opposite Directions
Scenario:
- Plane A flies east at 300 km/h
- Plane B flies west at 350 km/h
Setup: Take east as the positive (+) direction.
- vA = +300 km/h (east)
- vB = −350 km/h (west is negative)
Find: Relative velocity of A with respect to B (vAB)
Solution:
vAB = vA − vB = 300 − (−350) = 300 + 350 = +650 km/h
Interpretation: From B's viewpoint, A appears to approach at 650 km/h to the east. This large speed comes from them moving toward each other.
Check:
vBA = vB − vA = −350 − 300 = −650 km/h
From A's perspective, B appears to move at 650 km/h to the west.
Example 3: Chaining Velocities (Three Objects)
Scenario:
- Car A moves east at 300 km/h (relative to Earth)
- Car C moves at 100 km/h relative to Car A
- Both move in the same direction (east)
Find: Velocity of C relative to Earth
Solution: Use the formula rearranged:
vC = vCA + vA = 100 + 300 = 400 km/h
Interpretation: Although C moves at 100 km/h relative to A, relative to Earth, it moves at 400 km/h east because A itself is already moving at 300 km/h.
Maharashtra State Board: Class 11
Real-Life Examples
Two Cyclists
When one cyclist passes another on a narrow path:
- Actual speeds: Both may ride at 15 km/h
- Relative speed: You feel one approach at 30 km/h (when coming from opposite directions) or pass at 0 km/h (if from behind at the same speed)
Walking Inside a Moving Train or Bus
If a train moves east at 40 km/h and you walk east inside at 5 km/h:
- Relative to passengers on the train: You walk at 5 km/h
- Relative to observer on platform: You move at 40 + 5 = 45 km/h east
If you walk west (backward) at 5 km/h:
-
Relative to observer on platform: You move at 40 − 5 = 35 km/h east
Boat Crossing a River
A boat aims to cross a river perpendicular to the banks. The river current moves at 3 km/h.
- Boat's velocity relative to water: 10 km/h (straight across)
- Water's velocity relative to Earth: 3 km/h (downstream)
- Boat's velocity relative to Earth: Combination of both (resultant velocity)
The boat drifts downstream due to the current, even though it aims straight across.
Test Yourself
Shaalaa.com | Straight line motion Part 19 (Relative velocity)
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