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### Chapter 1 Unit(s) / Mechanics / Sig-Figs / Vectors

 Speed = (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 cm3 Sig Figs π = 3.14 (3 sigfig) π = 3.14159 (6 sigfig) Density = (mass / volume) || (kg / m3) || (g / cm3) √ = 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)Î2 + (y)ĵ2

 x = (-b +/- √b2 – 4 ⋅ a ⋅ c ) / (2 ⋅ a)

### Chapter 2: Motion along A Straight Line

 One 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 = ( (a-final) – (a-initial) ) / ( (t-final)-(t-initial) ) ∆t = (t-final) – (t-initial) ∆x = (x-final) – (x-initial) Acceleration ∆V = (V-final) – (V-initial) ∆t = (t-final) – (t-initial) 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 `V-final` = (a ⋅ t) + V-initial `V-final` 2 = (v-initial) 2 + 2 ⋅ a ( (t-final) – (t-initial) ) ∆x = (x-final) – (x-initial) ∆x = (v-average) ⋅ (seconds) ∆x = (1/2 ⋅ (V-final) + (V-initial) ) ⋅ t (seconds) `x-final` = 1/2 ( (V-initial) + (V-final) ) ⋅ t + (x-initial) `x-final` = x-initial + (V-initial) ⋅ t(seconds) + 1/2 ⋅ a ⋅ t 2 Gravity (g) = -9.8 m/s2 V-final = (V-initial) + g * t (seconds)

### Chapter 3: 2D or 3D Motion

 The Acceleration Vector a = ∆V / ∆t (v-final) = (v-initial) + ∆V ∆V = (v-final) – (v-initial) ∆V = (v-final) + (-(v-initial)) Constant Speed Changing Direction a = ∆V / ∆t (v-final) = (v-initial) + ∆V ∆V = (v-final) – (v-initial) Projectile Motion two assumptions: 1. The freefall acceleration (g) is constant 2. Air resistance is negligible y-direction = constant acceleration motion x-direction = constant velocity motion Acceleration is only negative (y-direction) g = -9.8 m/s2 Constant Velocity Motion x = (x-initial) + (v [x-direction] ) ⋅ t V (y-direction) = (v-initial) [y-direction] + g ⋅ t (y-final) = (y-initial + (v-initial) [y-direction] ⋅ t + 1/2 ⋅ g ⋅ t2 V (y-direction)2 = (v-initial) [y-direction]2 + 2 ⋅ g ( (y-final) – (y-initial) ) V (y-direction) = (v-initial) [y-direction] + g ⋅ t Trig Identity sin(ΘΘ) = sinΘcosΘ + cosΘsinΘ Constant Speed Motion velocity is always changing r = radius V = (2πr)2 : 4π2r T = time-period a = ∆V / ∆t : never zero ∆V = (V / r) · ∆r Centripetal Acceleration Ac = (V2) / r Ac = (2πr)2 / r Ac = 4π2r / T2 Tangential and Radial Acceleration Ac = a-rad Vector A-total = Vector A-tangential + Vector A-radical A-total = √(A-tan)2 + (A-rad)2 Relative Motion r ‘ = ( (v-initial) ⋅ t ) – (vector-r) Vector-r = √( (v-initial) ⋅ t)2 + (r ‘)2 Vector-V ‘ = (v-final) – (v-initial)

### Chapter 4: Newtons Laws

 Superposition of Forces Vector-R = Vector-F1 + Vector-F2 N = Net Force Fx = N · cos(ϴ) Fy = N · sin(ϴ) Rx = ∑Fx Ry = ∑Fy R = √(Rx)2 + Ry2 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 (x-direction) = (Fx total) / mass a (y-direction) = (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 Laws

 vector-F = 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 v2 Uniform Circular Motion Fc = m · ac : m · V2 / r Vertical Circle Top: Fy = -m · (V2 / r) Bottom: Fy = μs * m · (g + V2 / r) maxV = √(fs · r) / m maxV = √ μs · g · r Top View T · sin(ϴ) = m · ac ac = tan(ϴ) · g

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