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  Step Derivation/Formula Reasoning
  1 \Delta x = v_i t + \frac{1}{2}at^2 Use the equation of motion that relates initial velocity, acceleration, time, and displacement.
  2 \Delta x = (5.0 \, \text{m/s})(6.0 \, \text{s}) + \frac{1}{2}(2.0 \, \text{m/s}^2)(6.0 \, \text{s})^2 Substitute the given values: initial velocity v_i = 5.0 \, \text{m/s}, acceleration a = 2.0 \, \text{m/s}^2, and time t = 6.0 \, \text{s}.
  3 \Delta x = (5.0 \times 6.0) + \frac{1}{2} \times 2.0 \times (6.0)^2 Simplify the units and perform the necessary operations within the equation.
  4 \Delta x = 30.0 + \frac{1}{2} \times 2.0 \times 36.0 Calculate the product of initial velocity and time, and the square of time in the acceleration term.
  5 \Delta x = 30.0 + 1.0 \times 36.0 Compute the multiplication terms in the second part of the equation.
  6 \Delta x = 30.0 + 36.0 Add the two terms to get the total displacement.
  7 \Delta x = 66.0 \, \text{m} The total displacement during the first 6.0 seconds is \boxed{66.0} \, \text{meters}.
  
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  KinematicsForces
  \Delta x = v_i t + \frac{1}{2} at^2F = ma
  v = v_i + atF_g = \frac{G m_1m_2}{r^2}
  a = \frac{\Delta v}{\Delta t}f = \mu N
  R = \frac{v_i^2 \sin(2\theta)}{g}
  Circular MotionEnergy
  F_c = \frac{mv^2}{r}KE = \frac{1}{2} mv^2
  a_c = \frac{v^2}{r}PE = mgh
  KE_i + PE_i = KE_f + PE_f
  MomentumTorque and Rotations
  p = m v\tau = r \cdot F \cdot \sin(\theta)
  J = \Delta pI = \sum mr^2
  p_i = p_fL = I \cdot \omega
  Simple Harmonic Motion
  F = -k x
  T = 2\pi \sqrt{\frac{l}{g}}
  T = 2\pi \sqrt{\frac{m}{k}}
  ConstantDescription
  gAcceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface
  GUniversal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2
  \mu_k and \mu_sCoefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion.
  kSpring constant, in \text{N/m}
  M_E = 5.972 \times 10^{24} , \text{kg} Mass of the Earth
  M_M = 7.348 \times 10^{22} , \text{kg} Mass of the Moon
  M_M = 1.989 \times 10^{30} , \text{kg} Mass of the Sun
  VariableSI Unit
  s (Displacement)\text{meters (m)}
  v (Velocity)\text{meters per second (m/s)}
  a (Acceleration)\text{meters per second squared (m/s}^2\text{)}
  t (Time)\text{seconds (s)}
  m (Mass)\text{kilograms (kg)}
  VariableDerived SI Unit
  F (Force)\text{newtons (N)}
  E, PE, KE (Energy, Potential Energy, Kinetic Energy)\text{joules (J)}
  P (Power)\text{watts (W)}
  p (Momentum)\text{kilogram meters per second (kgm/s)}
  \tau (Torque)\text{newton meters (Nm)}
  I (Moment of Inertia)\text{kilogram meter squared (kgm}^2\text{)}
  f (Frequency)\text{hertz (Hz)}
  
  General Metric Conversion Chart
  
  Example of using unit analysis: Convert 5 kilometers to millimeters.
  
  1. Start with the given measurement: \text{5 km}
  
  2. Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}
  
  3. Perform the multiplication: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}
  
  4. Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}
  
  Prefix
  
  Symbol
  
  Power of Ten
  
  Equivalent
  
  Pico-
  
  p
  
  10^{-12}
  
  Nano-
  
  n
  
  10^{-9}
  
  Micro-
  
  µ
  
  10^{-6}
  
  Milli-
  
  m
  
  10^{-3}
  
  Centi-
  
  c
  
  10^{-2}
  
  Deci-
  
  d
  
  10^{-1}
  
  (Base unit)
  
  10^{0}
  
  Deca- or Deka-
  
  da
  
  10^{1}
  
  Hecto-
  
  h
  
  10^{2}
  
  Kilo-
  
  k
  
  10^{3}
  
  Mega-
  
  M
  
  10^{6}
  
  Giga-
  
  G
  
  10^{9}
  
  Tera-
  
  T
  
  10^{12}
  
  1. Some answers may be slightly off by 1% depending on rounding, etc.
  2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
  3. Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
  4. Bookmark questions that you can’t solve so you can come back to them later.
  5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!
  
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