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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

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 = ( (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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