Kinematics Cheatsheet

Cheatsheet Content

### Basic Definitions - **Mechanics:** Study of motion. - **Kinematics:** Describes motion without considering its causes (forces). - **Dynamics:** Explains motion by considering its causes (forces). - **Particle:** An object whose size and shape are negligible. - **Reference Frame:** A coordinate system relative to which motion is described. #### Scalar vs. Vector Quantities - **Scalar:** Quantity with magnitude only (e.g., distance, speed, time, mass). - **Vector:** Quantity with both magnitude and direction (e.g., displacement, velocity, acceleration, force). ### Distance and Displacement - **Distance ($d$):** Total path length covered by an object. - Scalar quantity. - Always positive. - **Displacement ($\Delta \vec{x}$ or $\vec{s}$):** Change in position of an object. - Vector quantity. - $\Delta \vec{x} = \vec{x}_{\text{final}} - \vec{x}_{\text{initial}}$ - Can be positive, negative, or zero. - **Relationship:** Distance $\ge$ $|\text{Displacement}|$ ### Speed and Velocity - **Speed ($v$):** Rate at which distance is covered. - Scalar quantity. - Average Speed: $v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{\Delta t}$ - Instantaneous Speed: Magnitude of instantaneous velocity. - **Velocity ($\vec{v}$):** Rate of change of displacement. - Vector quantity. - Average Velocity: $\vec{v}_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{\Delta \vec{x}}{\Delta t}$ - Instantaneous Velocity: $\vec{v} = \frac{d\vec{x}}{dt}$ (derivative of position with respect to time). - **Units:** meters per second (m/s) in SI. ### Acceleration - **Acceleration ($\vec{a}$):** Rate of change of velocity. - Vector quantity. - Average Acceleration: $\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_{\text{final}} - \vec{v}_{\text{initial}}}{\Delta t}$ - Instantaneous Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}$ (derivative of velocity, second derivative of position). - **Units:** meters per second squared (m/s$^2$) in SI. - **Interpretation:** - If $\vec{a}$ and $\vec{v}$ are in the same direction, speed increases. - If $\vec{a}$ and $\vec{v}$ are in opposite directions, speed decreases (deceleration). - If $\vec{a}$ is perpendicular to $\vec{v}$, direction changes (e.g., circular motion), but speed may be constant. ### Equations of Motion (Constant Acceleration) Valid only when acceleration ($\vec{a}$) is constant. 1. **Velocity-Time Relation:** $$\vec{v} = \vec{u} + \vec{a}t$$ (where $\vec{u}$ is initial velocity, $\vec{v}$ is final velocity) 2. **Displacement-Time Relation:** $$\vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2$$ 3. **Velocity-Displacement Relation:** $$v^2 = u^2 + 2\vec{a} \cdot \vec{s}$$ (Note: This is a scalar equation for 1D motion, or dot product for 3D) 4. **Displacement (alternative):** $$\vec{s} = \frac{(\vec{u} + \vec{v})}{2}t$$ 5. **Displacement in n-th second:** $$s_n = \vec{u} + \frac{\vec{a}}{2}(2n - 1)$$ **Graphical Interpretation:** - **Position-Time Graph ($\vec{x}$ vs. $t$):** - Slope = Instantaneous Velocity. - Concavity indicates acceleration. - **Velocity-Time Graph ($\vec{v}$ vs. $t$):** - Slope = Instantaneous Acceleration. - Area under curve = Displacement. - **Acceleration-Time Graph ($\vec{a}$ vs. $t$):** - Area under curve = Change in Velocity. ### Free Fall (Motion under Gravity) - In the absence of air resistance, all objects fall with the same constant acceleration. - **Acceleration due to gravity ($g$):** - On Earth's surface, $g \approx 9.8 \text{ m/s}^2$ (downwards). - Often approximated as $10 \text{ m/s}^2$ for simpler calculations. - Use the equations of motion by replacing $\vec{a}$ with $\vec{g}$. - Common convention: Upwards is positive, downwards is negative. So $\vec{a} = -g$. **Key Points:** - At the maximum height, vertical velocity ($v_y$) is zero. - Time of ascent = Time of descent (if starting and ending at same height). - Speed just before hitting the ground = Initial speed (if launched from and returning to same height). ### Projectile Motion - Motion of an object launched into the air, subject only to gravity. - **Assumptions:** Negligible air resistance, constant $g$. - **Key Idea:** Horizontal and vertical motions are independent. #### Horizontal Motion - Constant velocity ($v_x$). - No acceleration ($a_x = 0$). - Equation: $x = v_{x_0} t$ #### Vertical Motion - Constant acceleration ($a_y = -g$). - Equations (using $u_y$ as initial vertical velocity): - $v_y = u_y - gt$ - $y = u_y t - \frac{1}{2}gt^2$ - $v_y^2 = u_y^2 - 2gy$ #### Combined Motion - **Initial Velocity ($\vec{u}$):** If launched at angle $\theta$ with speed $u$: - $u_x = u \cos\theta$ - $u_y = u \sin\theta$ - **Time of Flight ($T$):** Time projectile remains in air. - $T = \frac{2u \sin\theta}{g}$ (for projectile starting and ending at same height) - **Maximum Height ($H$):** Highest point reached. - $H = \frac{u^2 \sin^2\theta}{2g}$ - **Horizontal Range ($R$):** Total horizontal distance covered. - $R = \frac{u^2 \sin(2\theta)}{g}$ - Maximum range at $\theta = 45^\circ$. - **Trajectory Equation:** $y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$ ### Relative Motion - Velocity of object A relative to object B: $$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$ - If we want velocity of B relative to A: $$\vec{v}_{BA} = \vec{v}_B - \vec{v}_A = -\vec{v}_{AB}$$ - **Example:** A person walking on a moving train. - $\vec{v}_{\text{person/ground}} = \vec{v}_{\text{person/train}} + \vec{v}_{\text{train/ground}}$ #### River-Boat Problems - **$\vec{v}_r$**: Velocity of river current. - **$\vec{v}_b$**: Velocity of boat in still water (its own speed). - **$\vec{v}_{br}$**: Velocity of boat relative to river. - **$\vec{v}_{bg}$**: Velocity of boat relative to ground (actual velocity). - **Vector Sum:** $\vec{v}_{bg} = \vec{v}_b + \vec{v}_r$ **Case 1: Crossing the river in shortest time** - Boat heads perpendicular to the current. - Time: $t = \frac{\text{width}}{v_b}$ - Drift: $x = v_r \cdot t = \frac{v_r \cdot \text{width}}{v_b}$ **Case 2: Crossing the river with minimum drift** - Boat heads upstream at an angle such that $\vec{v}_{bg}$ is perpendicular to $\vec{v}_r$. - $\sin\theta = \frac{v_r}{v_b}$ (where $\theta$ is angle with perpendicular to current). - Time: $t = \frac{\text{width}}{\sqrt{v_b^2 - v_r^2}}$ (only possible if $v_b > v_r$) #### Rain-Man Problems - $\vec{v}_m$: Velocity of man. - $\vec{v}_r$: Velocity of rain (relative to ground). - $\vec{v}_{rm}$: Velocity of rain relative to man. $$\vec{v}_{rm} = \vec{v}_r - \vec{v}_m$$ The direction of $\vec{v}_{rm}$ is the direction in which the man perceives the rain to fall.