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Chapter 1 Unit(s) / Mechanics / SigFigs / 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 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 Formula
x = (b +/ √b^{2}  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 = ( (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 Motion
The 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 Laws
Superposition 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 Laws
vectorF = 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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