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Chapter 1 Unit(s) / Mechanics / SigFigs / VectorsSpeed = (d/t)  (m/s)  d = distance : m = meters   t = time : s = seconds  1 km = 1000 m  1 kg = 1000 g  mass = (kg)  1 hour = 3600 seconds  time = (seconds)  1 mile = 1.609 km  length = (meter)  Volume = 1 cm^{3}  Sig Figs  π = 3.14 (3 sigfig)  π = 3.14159 (6 sigfig)  Density = (mass / volume)  (kg / m^{3})  (g / cm^{3})  √ = square root  Vector (Displacement) = √(x)^{2}+(y)^{2}  Total distance = x + y  Vector A = Vector B if Vector A = Vector B  Magnitude: √(x)^{2}+(y)^{2} = (Answer in Units) : 1 Direction  Components of Vector  Vector A = Ax + Ay  Ax = A ⋅ cos(Θ) Ay = A ⋅ sin(Θ)  A = √(A ⋅ cos(Θ))^{2} + (A ⋅ sin(Θ))^{2}  Θ = Angle  x = cos(Θ) y = sin(Θ)  cos(Θ) = Ax / A sin(Θ) = Ay / A  tan(Θ) = (y / x) or (Ay / Ax) or (By / Bx)  x = Î y = ĵ z = k̂  Vector A = AxÎ + Ayĵ Vector B = BxÎ + Byĵ  Vector R = Vector A + Vector B Vector R = (Ax + Bx)Î + (Ay + By)ĵ Vector R (direction) = (x)Î + (y)ĵ Vector R (magnitude) = √(x)Î^{2} + (y)ĵ^{2} 
Quadratic Formulax = (b +/ √b^{2}  4 ⋅ a ⋅ c ) / (2 ⋅ a) 
  Chapter 2: Motion along A Straight LineOne Dimensional Motion  Average Speed = (total distance) / (time)
 Displacement = Final Point  Initial Point
 Not Constant Velocity  Average Velocity (V) = (displacement / time)
 Average Velocity (V) = (∆x / ∆t)
 Instantaneous Velocity = derivative of the given equation Instantaneous Velocity = ( (afinal)  (ainitial) ) / ( (tfinal)(tinitial) )  ∆t = (tfinal)  (tinitial)  ∆x = (xfinal)  (xinitial)  Acceleration  ∆V = (Vfinal)  (Vinitial) ∆t = (tfinal)  (tinitial)  Acceleration (a) = (∆V) / (∆t) [a is constant]  if a > 0 (positive) if a < 0 (negative)  Instantaneous Acceleration = derivative of the given equation  Constant Acceleration = constant acceleration motion in 1D  Vfinal = (a ⋅ t) + Vinitial Vfinal ^{2} = (vinitial)^{2} + 2 ⋅ a ( (tfinal)  (tinitial) )
 ∆x = (xfinal)  (xinitial) ∆x = (vaverage) ⋅ (seconds) ∆x = (1/2 ⋅ (Vfinal) + (Vinitial) ) ⋅ t (seconds)  xfinal = 1/2 ( (Vinitial) + (Vfinal) ) ⋅ t + (xinitial) xfinal = xinitial + (Vinitial) ⋅ t(seconds) + 1/2 ⋅ a ⋅ t^{2}
 Gravity (g) = 9.8 m/s^{2}  Vfinal = (Vinitial) + g * t (seconds) 
  Chapter 3: 2D or 3D MotionThe Acceleration Vector  a = ∆V / ∆t  (vfinal) = (vinitial) + ∆V ∆V = (vfinal)  (vinitial) ∆V = (vfinal) + ((vinitial))  Constant Speed Changing Direction  a = ∆V / ∆t  (vfinal) = (vinitial) + ∆V ∆V = (vfinal)  (vinitial)  Projectile Motion two assumptions:  1. The freefall acceleration (g) is constant 2. Air resistance is negligible  ydirection = constant acceleration motion xdirection = constant velocity motion Acceleration is only negative (ydirection) g = 9.8 m/s^{2}  Constant Velocity Motion  x = (xinitial) + (v [xdirection] ) ⋅ t  V (ydirection) = (vinitial) [ydirection] + g ⋅ t  (yfinal) = (yinitial + (vinitial) [ydirection] ⋅ t + 1/2 ⋅ g ⋅ t^{2}  V (ydirection)^{2} = (vinitial) [ydirection]^{2} + 2 ⋅ g ( (yfinal)  (yinitial) )  V (ydirection) = (vinitial) [ydirection] + g ⋅ t  Trig Identity  sin(ΘΘ) = sinΘcosΘ + cosΘsinΘ  Constant Speed Motion velocity is always changing  r = radius  V = (2πr)^{2} : 4π^{2}r  T = timeperiod  a = ∆V / ∆t : never zero ∆V = (V / r) · ∆r  Centripetal Acceleration  Ac = (V^{2}) / r Ac = (2πr)^{2} / r Ac = 4π^{2}r / T^{2}  Tangential and Radial Acceleration  Ac = arad  Vector Atotal = Vector Atangential + Vector Aradical Atotal = √(Atan)^{2} + (Arad)^{2}  Relative Motion  r ' = ( (vinitial) ⋅ t )  (vectorr)  Vectorr = √( (vinitial) ⋅ t)^{2} + (r ')^{2}  VectorV ' = (vfinal)  (vinitial) 
  Chapter 4: Newtons LawsSuperposition of Forces  VectorR = VectorF1 + VectorF2  N = Net Force  Fx = N · cos(ϴ) Fy = N · sin(ϴ)  Rx = ∑Fx Ry = ∑Fy  R = √(Rx)^{2} + Ry^{2}  Newton's 1st Law No Force; No Acceleration; No Motion  Inertia: the tendency of an object to resist any attempt to change its velocity  Newton's 2nd Law  Net Force = m · g  a (xdirection) = (Fx total) / mass a (ydirection) = (Fy total) / mass  tan(ϴ) = y / x  Newton's 3rd Law  Fn = Normal Force  Fy = Fn  m · g · cos(ϴ)  Fx = m · g · sin(ϴ) 
Chapter 5: Applying Newton's LawsvectorF = m · a  Fx = m · ax  T = tension : friction  Fy= m · ay  y = T  m · g  Fr = Fn : Normal Force (Fn)  No Friction  α = Coefficient  Fn = m · g  Fx = T1· cos(ϴ) + T2· cos(ϴ)   Fy = T1· sin(ϴ) + T2· sin(ϴ)  Friction  Static Friction (fs): Object not in motion Kinetic Friction (fK): Object is in motion   Empirical Formula  μk: Coefficient of Kinetic Friction μs: Coefficient of Static Friction Static: fs ≤ μs · Fn Static: fk = μk · Fn   Terminal Speed  Fr α v   Fr α v^{2}  Uniform Circular Motion  Fc = m · ac : m · V^{2} / r  Vertical Circle  Top: Fy = m · (V^{2} / r) Bottom: Fy = μs * m · (g + V^{2} / r) maxV = √(fs · r) / m maxV = √ μs · g · r  Top View  T · sin(ϴ) = m · ac ac = tan(ϴ) · g 

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