Vectors and Motion

Vectors

• Vectors are represented as $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$

Sum of Vectors

• The sum of two vectors $\vec{A}$ and $\vec{B}$ is calculated by adding their corresponding components: $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$

Scalar Product

• The scalar product (dot product) of two vectors is $\vec{A} \cdot \vec{B} = |A||B|cos(\theta)$
• Can also be computed by $\vec{A} \cdot \vec{B} = (A_x)(B_x) + (A_y)(B_y) + (A_z)(B_z)$

Vector Product

• The vector product (cross product) is $\vec{A} \times \vec{B} = |A||B|sen(\theta)\hat{n}$
• The vector product can be calculated using the determinant of a matrix formed by the components of the vectors: $\vec{A} \times \vec{B} = det\begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{pmatrix}$

Kinematics: MRU (Uniform Rectilinear Motion)

• Position as a function of time is given by $x = x_0 + v(t - t_0)$

Kinematics: MRUV (Uniformly Accelerated Rectilinear Motion)

• Position is $x = x_0 + v_0(t - t_0) + \frac{1}{2}a(t - t_0)^2$
• Velocity is $v = v_0 + a(t - t_0)$
• Velocity squared $v^2 = v_0^2 + 2a(x - x_0)$

Vertical Throw and Free Fall

• Vertical position is $y = y_0 + v_0(t - t_0) - \frac{1}{2}g(t - t_0)^2$
• Vertical speed is $v = v_0 - g(t - t_0)$
• Velocity squared is $v^2 = v_0^2 - 2g(y - y_0)$

Oblique Throw

• Horizontal acceleration $a_x = 0$ and vertical acceleration $a_y = -g$
• Initial horizontal velocity $v_{0x} = v_0cos(\alpha)$
• Initial vertical velocity $v_{0y} = v_0sen(\alpha)$
• Horizontal position $x = x_0 + v_{0x}(t - t_0)$
• Vertical position is $y = y_0 + v_{0y}(t - t_0) - \frac{1}{2}g(t - t_0)^2$
• Horizontal velocity $v_x = v_{0x}$
• Vertical velocity $v_y = v_{0y} - g(t - t_0)$

Uniform Circular Motion (MCU)

• Angular velocity $\omega = \frac{\Delta \theta}{\Delta t}$
• Tangential speed $v = \omega r$
• Centripetal acceleration $a_c = \frac{v^2}{r} = \omega^2 r$

Uniformly Accelerated Circular Motion (MCUV)

• Angular acceleration $\alpha = \frac{\Delta \omega}{\Delta t}$
• Angular position $\theta = \theta_0 + \omega_0(t - t_0) + \frac{1}{2}\alpha(t - t_0)^2$
• Angular velocity $\omega = \omega_0 + \alpha(t - t_0)$
• Angular velocity squared $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Dynamics: Newton's Laws

• $\sum \vec{F} = m\vec{a}$
• $F_{AB} = -F_{BA}$

Friction

• Static friction $f_s \leq \mu_s N$
• Kinetic friction $f_k = \mu_k N$

Work and Energy

• Work done $W = \vec{F} \cdot \vec{d} = |F||d|cos(\theta)$
• Net work done $W_{neto} = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
• Gravitational potential energy $U_g = mgy$
• Elastic potential energy $U_e = \frac{1}{2}kx^2$
• Mechanical energy $E_{mec} = K + U$
• Change in mechanical energy $\Delta E_{mec} = W_{NC}$
• Power $P = \frac{W}{\Delta t} = \vec{F} \cdot \vec{v}$

Impulse and Momentum

• Impulse $\vec{I} = \int \vec{F} dt = \Delta \vec{p}$
• Momentum $\vec{p} = m\vec{v}$
• Conservation of momentum $\sum \vec{p_i} = \sum \vec{p_f}$

Collisions

• Elastic collisions have a coefficient of restitution $e = 1$ and kinetic energy is conserved $K_i = K_f$
• Inelastic collisions have $0 < e < 1$ and $K_i > K_f$
• Plastic collisions have $e = 0$, $K_i > K_f$, and the objects stick together, $v_{1f} = v_{2f}$

Coefficient of Restitution

• $e = - \frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$

Statics

• Equilibrium conditions $\sum \vec{F} = 0$ and $\sum \vec{\tau} = 0$
• Torque $\tau = rFsen(\theta)$