Concept Card: Relative Velocity in Two Dimensions (River-Boat & Rain-Man) 1. Fundamental Formulation of 2D Relative Velocity: The velocity of object $A$ observed from the frame of reference of object $B$ is given by vector subtraction: $\vec{v}_{AB} = \vec{v}_A – \vec{v}_B$ $\vec{v}_A = \vec{v}_{AB} + \vec{v}_B$ where $\vec{v}_A$ and $\vec{v}_B$ are measured relative to a … Read more
Concept Card: Vectors & Vector Algebra (Addition & Subtraction) 1. Scalars vs. Vectors: Scalar Quantity: A physical quantity completely specified by a numerical magnitude (number + unit) and sign. It obeys the ordinary rules of algebra. Examples: Mass, time, distance, speed, work, energy, temperature, electric charge. Vector Quantity: A physical quantity possessing both magnitude and … Read more
Concept Card: Master Review & Strategy Sheet for 1D Kinematics 1. The Calculus Backbone of Kinematics: When acceleration is non-constant, kinematic formulas cannot be applied directly; calculus definitions must be used: Velocity: $v(t) = \frac{dx}{dt} \iff x(t) = x_0 + \int_{0}^{t} v(t’)\,dt’$ Acceleration (Time Form): $a(t) = \frac{dv}{dt} = \frac{d^2 x}{dt^2} \iff v(t) = u … Read more
Concept Card: Relative Velocity in One Dimension (1D Kinematics) 1. Definition of Relative Velocity: The velocity of a body $A$ as observed from the reference frame of another body $B$ is defined as the relative velocity of $A$ with respect to $B$ ($\vec{v}_{AB}$): $\vec{v}_{AB} = \vec{v}_A – \vec{v}_B$ Similarly, the velocity of $B$ relative to … Read more
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