Chapter 1 - 5 Formulas
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Speed = (d/t) || (m/s)
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d = distance : m = meters
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t = time : s = seconds
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1 km = 1000 m
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1 kg = 1000 g
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mass = (kg)
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1 hour = 3600 seconds
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time = (seconds)
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1 mile = 1.609 km
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length = (meter)
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Volume = 1 cm3
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Sig Figs
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π = 3.14 (3 sigfig)
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π = 3.14159 (6 sigfig)
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Density = (mass / volume) || (kg / m3) || (g / cm3)
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√ = square root
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Vector (Displacement) = √(x)2+(y)2
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Total distance = x + y
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Vector A = Vector B if |Vector A| = |Vector B|
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Magnitude: √(x)2+(y)2 = (Answer in Units) : 1 Direction
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Components of Vector
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Vector A = Ax + Ay
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Ax = A ⋅ cos(Θ)
Ay = A ⋅ sin(Θ) |
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A = √(A ⋅ cos(Θ))2 + (A ⋅ sin(Θ))2
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Θ = Angle
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x = cos(Θ)
y = sin(Θ) |
cos(Θ) = Ax / A
sin(Θ) = Ay / A |
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tan(Θ) = (y / x) or (Ay / Ax) or (By / Bx)
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x = Î
y = ĵ z = k̂ |
Vector A = AxÎ + Ayĵ
Vector B = BxÎ + Byĵ |
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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 |
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x = (-b +/- √b2 - 4 ⋅ a ⋅ c ) / (2 ⋅ a)
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One Dimensional Motion
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Average Speed = (total distance) / (time) |
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Displacement = Final Point - Initial Point |
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Not Constant Velocity
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Average Velocity (V) = (displacement / time) |
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Average Velocity (V) = (∆x / ∆t) |
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Instantaneous Velocity = derivative of the given equation
Instantaneous Velocity = ( (a-final) - (a-initial) ) / ( (t-final)-(t-initial) ) |
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∆t = (t-final) - (t-initial)
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∆x = (x-final) - (x-initial)
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Acceleration
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∆V = (V-final) - (V-initial)
∆t = (t-final) - (t-initial) |
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Acceleration (a) = (∆V) / (∆t)
[a is constant] |
if a > 0 (positive)
if a < 0 (negative) |
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Instantaneous Acceleration = derivative of the given equation
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Constant Acceleration
= constant acceleration motion in 1D |
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V-final = (a ⋅ t) + V-initial V-final2 = (v-initial)2 + 2 ⋅ a ( (t-final) - (t-initial) ) |
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∆x = (x-final) - (x-initial)
∆x = (v-average) ⋅ (seconds) ∆x = (1/2 ⋅ (V-final) + (V-initial) ) ⋅ t (seconds) |
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x-final = 1/2 ( (V-initial) + (V-final) ) ⋅ t + (x-initial) x-final = x-initial + (V-initial) ⋅ t(seconds) + 1/2 ⋅ a ⋅ t2 |
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Gravity (g) = -9.8 m/s2
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V-final = (V-initial) + g * t (seconds)
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The Acceleration Vector
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a = ∆V / ∆t
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(v-final) = (v-initial) + ∆V
∆V = (v-final) - (v-initial) ∆V = (v-final) + (-(v-initial)) |
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Constant Speed Changing Direction
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a = ∆V / ∆t
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(v-final) = (v-initial) + ∆V
∆V = (v-final) - (v-initial) |
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Projectile Motion
two assumptions: |
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1. The freefall acceleration (g) is constant
2. Air resistance is negligible |
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y-direction = constant acceleration motion
x-direction = constant velocity motion Acceleration is only negative (y-direction) g = -9.8 m/s2 |
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Constant Velocity Motion
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x = (x-initial) + (v [x-direction] ) ⋅ t
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V (y-direction) = (v-initial) [y-direction] + g ⋅ t
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(y-final) = (y-initial + (v-initial) [y-direction] ⋅ t + 1/2 ⋅ g ⋅ t2
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V (y-direction)2 = (v-initial) [y-direction]2 + 2 ⋅ g ( (y-final) - (y-initial) )
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V (y-direction) = (v-initial) [y-direction] + g ⋅ t
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Trig Identity
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sin(ΘΘ) = sinΘcosΘ + cosΘsinΘ
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Constant Speed Motion
velocity is always changing |
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r = radius
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V = (2πr)2 : 4π2r
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T = time-period
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a = ∆V / ∆t : never zero
∆V = (V / r) · ∆r |
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Centripetal Acceleration
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Ac = (V2) / r
Ac = (2πr)2 / r Ac = 4π2r / T2 |
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Tangential and Radial Acceleration
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Ac = a-rad
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Vector A-total = Vector A-tangential + Vector A-radical
A-total = √(A-tan)2 + (A-rad)2 |
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Relative Motion
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r ' = ( (v-initial) ⋅ t ) - (vector-r)
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Vector-r = √( (v-initial) ⋅ t)2 + (r ')2
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Vector-V ' = (v-final) - (v-initial)
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Superposition of Forces
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Vector-R = Vector-F1 + Vector-F2
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N = Net Force
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Fx = N · cos(ϴ)
Fy = N · sin(ϴ) |
Rx = ∑Fx
Ry = ∑Fy |
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R = √(Rx)2 + Ry2
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Newton's 1st Law
No Force; No Acceleration; No Motion |
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Inertia:
the tendency of an object to resist any attempt to change its velocity |
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Newton's 2nd Law
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Net Force = m · g
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a (x-direction) = (Fx total) / mass
a (y-direction) = (Fy total) / mass |
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tan(ϴ) = y / x
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Newton's 3rd Law
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Fn = Normal Force
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Fy = Fn - m · g · cos(ϴ)
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Fx = m · g · sin(ϴ)
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vector-F = m · a
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Fx = m · ax
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T = tension : friction
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Fy= m · ay
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y = T - m · g
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Fr = Fn : Normal Force (Fn)
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No Friction
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α = Coefficient
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Fn = m · g
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Fx = T1· cos(ϴ) + T2· cos(ϴ)
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Fy = T1· sin(ϴ) + T2· sin(ϴ)
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Friction
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Static Friction (fs): Object not in motion
Kinetic Friction (fK): Object is in motion |
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Empirical Formula
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μk: Coefficient of Kinetic Friction
μs: Coefficient of Static Friction Static: fs ≤ μs · Fn Static: fk = μk · Fn |
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Terminal Speed
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Fr α v
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Fr α v2
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Uniform Circular Motion
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Fc = m · ac : m · V2 / r
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Vertical Circle
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Top: Fy = -m · (V2 / r)
Bottom: Fy = μs * m · (g + V2 / r) maxV = √(fs · r) / m maxV = √ μs · g · r |
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Top View
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T · sin(ϴ) = m · ac
ac = tan(ϴ) · g |
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