Concept Card: Motion Under Gravity & Vertical Kinematics 1. Governing Principles & Sign Conventions: Motion under gravity is a classic case of one-dimensional uniformly accelerated motion where acceleration is strictly constant near the surface of Earth: $\vec{a} = \vec{g} \approx 9.8\text{ m/s}^2 \quad (\text{often taken as } 10\text{ m/s}^2 \text{ in competitive problems})$ Taking the … Read more
Concept Card: Velocity-Time (v-t) Graphs & Graphical Kinematics 1. Physical Significance of Slope in a $v-t$ Graph: The slope of a velocity-time graph represents the time rate of change of velocity, which is acceleration. Instantaneous Acceleration ($a$): Given by the slope of the tangent line drawn to the $v-t$ curve at time $t$: $a = … Read more
Concept Card: Position-Time (x-t) Graphs & Motion Interpretation 1. Physical Meaning of the Slope of an $x-t$ Graph: The slope of a position-time graph represents the rate of change of position with respect to time, which is velocity. Instantaneous Velocity ($v$): The slope of the tangent line drawn to the $x-t$ curve at any instant … Read more
Concept Card: Third Equation of Motion & Kinematic Problem Solving 1. Mathematical Derivation of the Third Kinematic Equation ($v^2 = u^2 + 2as$): Calculus Derivation via Differential Chain Rule: Acceleration is defined as the time derivative of velocity: $a = \frac{dv}{dt}$. Applying the chain rule of differentiation with respect to spatial displacement $s$: $a = … Read more
Concept Card: Acceleration & Uniformly Accelerated Motion 1. Definition & Mathematical Properties of Acceleration: Acceleration ($\vec{a}$) is defined as the time rate of change of velocity: $\vec{a} = \frac{d\vec{v}}{dt}$ It is a vector quantity with dimensions $[M^0 L^1 T^{-2}]$ and SI unit $\text{m/s}^2$. Average Acceleration ($\vec{a}_{\text{avg}}$): $\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f – \vec{v}_i}{t_f … Read more
Concept Card: Speed, Velocity & Types of Motion 1. Speed vs. Velocity: Speed ($v$): The rate at which distance is covered with respect to time. It is a scalar quantity ($v \ge 0$) with dimensions $[L T^{-1}]$ and SI unit $\text{m/s}$. Velocity ($\vec{v}$): The rate of change of the position vector with respect to time … Read more
Concept Card: Frame of Reference, Distance & Displacement in 1D Motion 1. Frame of Reference: A frame of reference is a coordinate system (Cartesian axes $x, y, z$) attached to an observer along with a calibrated clock to record the time coordinate $t$. Inertial Frame of Reference: A frame in which Newton’s First Law of … Read more
Concept Card: Master Review & Strategy Sheet for Units, Errors & Dimensions 1. Dimensional Analysis Master Relations: The dimensions of any physical quantity $Q$ are represented as $[Q] = [M^a L^b T^c A^d K^e \text{mol}^f \text{cd}^g]$. Fundamental Electromagnetic Dimensionless Ratios: Fine-structure constant: $\alpha = \frac{e^2}{4\pi \varepsilon_0 \hbar c} \approx \frac{1}{137} \implies [M^0 L^0 T^0]$ Speed … Read more
Concept Card: Applications & Limitations of Dimensional Analysis 1. The Principle of Homogeneity of Dimensions: The Principle of Homogeneity states that a physical equation is dimensionally valid if and only if every term on both sides of the equation has the exact same dimensional formula. If an equation has the general form: $A = B […]
Concept Card: Dimensions of Physical Quantities & Dimensional Formulae 1. Definition of Dimensions: The dimensions of a physical quantity are the powers (or exponents) to which the fundamental base quantities must be raised to represent that quantity. The standard dimensional expression is represented in terms of the seven base dimensions: $[Q] = [M^a L^b T^c … Read more