Concept Card: Projectile Motion on an Inclined Plane & Uniform Circular Motion Part A: Projectile Motion on an Inclined Plane 1. Coordinate Frame Transformation: Orient the coordinate axes relative to the incline: – $+x’$ axis: Directed up along the surface of the incline. – $+y’$ axis: Directed perpendicular (normal) to the incline. Let the inclination … Read more
Concept Card: Oblique Projectile Motion 1. Physical Framework & Initial Conditions: A particle launched from ground level with initial speed $u$ at an angle of elevation $\theta$ ($0 < \theta < 90^\circ$) to the horizontal moves in a vertical plane under the constant downward acceleration of gravity $\vec{g}$. Taking the launch point as origin $O(0, ... Read more
Concept Card: Horizontal Projectile Motion 1. Physical Framework & Coordinate Setup: A projectile launched horizontally with initial speed $u$ from a height $h$ above level ground moves under the sole influence of gravity (neglecting air resistance). Choosing the point of projection as the origin $O(0, 0)$, with the horizontal forward direction as $+x$ and the … Read more
Concept Card: Motion in a Plane & 2D Kinematics 1. Position and Displacement Vectors in Two Dimensions (2D): In a Cartesian coordinate plane ($x-y$), the position of a particle at time $t$ is represented by the position vector $\vec{r}(t)$ directed from origin $O$ to point $P(x, y)$: Position Vector: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$ Magnitude: … Read more
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