Vectors and Relative Velocity

Vectors

• Vector addition can be done graphically using the "tip to tail" method, where order is irrelevant and the resultant's length and angle are measured.
• Alternatively, use the component method: resolve each vector into x and y components ($A_x = A\cos\theta$, $A_y = A\sin\theta$, using +/- signs for direction), add the x and y components to get Rx and Ry ($R_x = A_x + B_x + C_x +...$, $R_y = A_y + B_y + C_y +...$), and then find the resultant's magnitude and direction ($R = \sqrt{R_x^2 + R_y^2}$, $\theta = \arctan(\frac{R_y}{R_x})$, using +/- signs of $R_x$ & $R_y$ to determine the quadrant of $\theta$).
• For a vector pointing along the +x axis, $A_x = A$ and $A_y = 0$.
• For a vector pointing along the +y axis, $A_x = 0$ and $A_y = A$.

Relative Velocity

• $\vec{v}{object/ground} = \vec{v}{object/moving} + \vec{v}_{moving/ground}$
• Example relative velocity is a boat crossing a river: $\vec{v}{boat/shore} = \vec{v}{boat/water} + \vec{v}_{water/shore}$

Kinematics

Constant Velocity

• $v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{\Delta t}$

Constant Acceleration

• $v = v_0 + at$
• $x = x_0 + v_0t + \frac{1}{2}at^2$
• $v^2 = v_0^2 + 2a(x - x_0)$
• $x = x_0 + \frac{1}{2}(v + v_0)t$

Free Fall

• Constant acceleration equations apply with $a = g = 9.8 m/s^2$
• Choose a direction to be positive, and at the top of the trajectory, $v = 0$.

Projectile Motion

• $a_x = 0$
• $a_y = -g = -9.8 m/s^2$
• $v_{0x} = v_0 \cos \theta$
• $v_{0y} = v_0 \sin \theta$

Level Ground Projectile Motion

• Range: $R = \frac{v_0^2 \sin 2\theta}{g}$
• Max Height: $H = \frac{v_0^2 \sin^2 \theta}{2g}$

Dynamics

Newton's Laws

• An object in motion stays in motion, and an object at rest stays at rest, unless acted upon by a force.
• $\Sigma \vec{F} = m\vec{a}$
• For every action, there is an equal and opposite reaction.

Types of Forces

• Weight: $W = mg$
• Normal Force: Contact force perpendicular to the surface.
• Tension: Force exerted by a string or rope.
• Friction: Force opposing motion, either Static: $f_s \le \mu_s N$, or Kinetic: $f_k = \mu_k N$.

Inclined Planes

• $F_{g||} = mg \sin \theta$
• $F_{g\perp} = mg \cos \theta$

Uniform Circular Motion

Equations

• $v = \frac{2\pi r}{T}$
• $a_c = \frac{v^2}{r}$
• $F_c = ma_c = \frac{mv^2}{r}$

Notes

• Velocity is tangent to the circle.
• Acceleration and net force point towards the center of the circle.
• In uniform circular motion, speed is constant, but velocity changes due to changing direction.