}]]}]}]}] 30:T2c76,

### Motion in a Straight Line (Ch 3) - **Position, Path Length, Displacement:** - Position: Location of an object with respect to origin. - Path Length: Total distance covered. Scalar. - Displacement: Change in position ($\Delta x = x_2 - x_1$). Vector. - **Speed & Velocity:** - Average Speed: Total path length / Total time. Scalar. - Average Velocity: Displacement / Total time ($\vec{v}_{avg} = \Delta \vec{x} / \Delta t$). Vector. - Instantaneous Velocity: $\vec{v} = d\vec{x}/dt$. - **Acceleration:** - Average Acceleration: $\Delta \vec{v} / \Delta t$. - Instantaneous Acceleration: $\vec{a} = d\vec{v}/dt = d^2\vec{x}/dt^2$. - **Kinematic Equations (Constant Acceleration):** 1. $v = v_0 + at$ 2. $x = x_0 + v_0t + \frac{1}{2}at^2$ 3. $v^2 = v_0^2 + 2a(x - x_0)$ 4. $x_t = v_0 + \frac{a}{2}(2t - 1)$ (Displacement in $t$-th second) ### Motion in a Plane (Ch 4) - **Scalars & Vectors:** - Scalar: Magnitude only (e.g., mass, speed, distance). - Vector: Magnitude and direction (e.g., displacement, velocity, force). - **Vector Operations:** - **Addition (Triangle/Parallelogram Law):** $\vec{R} = \vec{A} + \vec{B}$ - Magnitude: $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$ - Direction: $\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$ - **Resolution of Vectors:** $\vec{A} = A_x\hat{i} + A_y\hat{j}$ - **Dot Product (Scalar Product):** $\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$ - **Cross Product (Vector Product):** $\vec{A} \times \vec{B} = (AB\sin\theta)\hat{n}$ - Magnitude: $|\vec{A} \times \vec{B}| = AB\sin\theta$ - Direction: Right-hand rule. - **Projectile Motion:** - **Horizontal Range:** $R = \frac{u^2\sin(2\theta)}{g}$ - **Maximum Height:** $H = \frac{u^2\sin^2\theta}{2g}$ - **Time of Flight:** $T = \frac{2u\sin\theta}{g}$ - **Uniform Circular Motion:** - **Angular Velocity:** $\omega = \frac{d\theta}{dt} = \frac{v}{r}$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{r} = \omega^2 r$ (directed towards center) - **Centripetal Force:** $F_c = m a_c = \frac{mv^2}{r}$ ### Laws of Motion (Ch 5) - **Newton's First Law (Law of Inertia):** An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. - **Newton's Second Law:** $\vec{F} = m\vec{a}$ - Force is proportional to the rate of change of momentum: $\vec{F} = \frac{d\vec{p}}{dt}$. - Momentum: $\vec{p} = m\vec{v}$. - Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$. - **Newton's Third Law:** To every action, there is an equal and opposite reaction. - **Conservation of Momentum:** In an isolated system, the total momentum remains constant. - $m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$. - **Common Forces:** - **Weight:** $W = mg$ - **Normal Force:** Perpendicular to surface. - **Friction:** Opposes relative motion. - Static Friction: $f_s \le \mu_s N$ - Kinetic Friction: $f_k = \mu_k N$ ($\mu_k < \mu_s$) - **Equilibrium of a Particle:** Net force on the particle is zero ($\sum \vec{F} = 0$). ### Work, Energy and Power (Ch 6) - **Work Done:** - Constant Force: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$. - Variable Force: $W = \int \vec{F} \cdot d\vec{r}$. - **Kinetic Energy:** $K = \frac{1}{2}mv^2$. - **Work-Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$. - **Potential Energy:** - Gravitational PE: $U_g = mgh$. - Spring PE: $U_s = \frac{1}{2}kx^2$. - **Conservative & N