Constants
$$a_{\text{gravity}} = -9.8\text{ m/s}^2$$
Prior Knowledge
- Trigonometric Ratios
- $$s\frac{o}{h} \hspace{2mm} c\frac{a}{h} \hspace{2mm} t\frac{o}{a}$$
- Pythagorean Theorem
- $$a^2 + b^2 = c^2$$
Unit Conversions
- $$1\text{ in} = 2.54\text{ cm}$$
When converting square or cubic units, remember to square or cube the conversion factor as well! $$1 m^3 = (100 cm)^3$$
Metric Prefixes
- G giga
- $$10^9$$
- M mega
- $$10^6$$
- k kilo
- $$10^3$$
- c centi
- $$10^{-2}$$
- m milli
- $$10^{-3}$$
- $$\mu$$ mc micro
- $$10^{-6}$$
- n nano
- $$10^{-9}$$
Vectors
- Directional properties with magnitude (absolute numerical value) and direction (angle).
- Vector components are the projection "shadow" of the vector onto the x- and y-axis.
Vector Addition and Subtraction
- Geometric Method
- Be able to add and subtract vectors by arranging them head to tail and connecting the first tail to the last head.
- Analytical Method
- Be able to add and subtract vectors by first taking their x- and y-components.
- Then add (or subtract) all of the x-components and separately add (or subtract) all of the y-components.
Vector Components
$$\tan\theta = \frac{A_y}{A_x}$$
$$A_x = A\cos\theta$$
$$A_y = A\sin\theta$$
$$\theta$$ is the angle going counterclockwise from the positive x-axis to the vector.
Vector Magnitude
$$A = \sqrt{A_x^2 + A_y^2 + A_z^2}$$
$$\tan\theta = \frac{A_y}{A_x}$$
Unit Vectors
- $$\hat{i}$$
- x-axis
- $$\hat{j}$$
- y-axis
- $$\hat{k}$$
- z-axis
One Dimensional Motion
- $$\Delta$$ Delta
- $$\Delta$$ means the change in a variable.
-
$$\Delta t = t - t_0$$
-
$$\Delta x = x - x_0$$
Kinematics
- Displacement Average
- $$\Delta x = x - x_0$$
- Velocity Average
- $$v = \frac{\Delta x}{\Delta t}$$
- Acceleration Average
- $$a = \frac{\Delta v}{\Delta t}$$
- Speed Average
- $$\text{speed} = \frac{\text{total distance}}{\Delta t}$$
- Velocity Instantaneous
- $$v = \frac{dx}{dt}$$
- Acceleration Instantaneous
- $$a = \frac{dv}{dt}$$
Derivatives
Don't forget the chain rule!
- $$[X^n]' = nx^{n-1}$$
- $$[\sin(kx)]' = k\cos(kx)$$
- $$[\cos(kx)]' = -k\sin(kx)$$
- $$[e^{kx}]' = ke^{kx}$$
Two-Dimensional Kinematics
- Use the x-kinematic equations to calculate the motion of the x-coordinate along the x-axis.
- Use the y-kinematic equations to trace the motion of the y-coordinate along the y-axis.
- X-Axis
- $$a_x = 0$$
- Y-Axis Free Fall
- $$a_y = -9.81\text{ m/s}^2$$
Relative Motion
/ Reference Frames
If an object moves with velocity $$\vec{v}_{o1}$$ with respect to reference frame (set of x-y axes) "1" and if reference frame "1" moves with velocity $$\vec{v}_{12}$$ with respect to reference frame "2", then the velocity of the object with respect to reference frame "2" will be given by the (vector) equation:
$$\boxed{\vec{v}_{o2} = \vec{v}_{o1} + \vec{v}_{12}}$$