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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 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

Quadratic Formula
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 = ( (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/s2

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/s2

Constant Velocity Motion

x = (xinitial) + (v [xdirection] ) ⋅ t

V (ydirection) = (vinitial) [ydirection] + g ⋅ t

(yfinal) = (yinitial + (vinitial) [ydirection] ⋅ t + 1/2 ⋅ g ⋅ t2

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 : 4π2r

T = timeperiod

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 = 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 + 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 (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 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

