Advanced Vector Calculus Flashcards
Principals of Trajectory Design
F
CORDIC Algorithms
S
Bezier Curves
D
Kepler’s Second Law in Vectors
Law of Equal Areas
The area inside the ellipse, the net space between the initial (a) and final (b) distances from the smaller to the larger mass, is equal to all other areas measured in the same length of time, regardless of the larger masses position within the ellipse.
Meaning the magnitude of the vector function at the final time (t) can be calculated through √(a^2sin^2(t)+b^2cos^2(t)). Integral of this is arc length. Derivative (acceleration) is proportional to the radius (r).
Kepler’s Third Law in Vectors
Law of Harmonies
Period of orbit (T) squared over average distance (d) cubed is known as the ‘orbital ratio’
Equal to those of similar average vectors
(T1^2)/(R1^3)≈(T2^2)/(R2^3)
Orbital Harmony
Orbital systems that are said to have similar average vector magnitudes to one another
Derivative Vector
Limit of the position vector function between the vector r(t) from center to a point on a curve and another vector r(t+Δt) that describes that distance after a certain time period as Δt approaches 0
r’(t)=Lim(Δt→0) (r(t+Δt)-r(t))/Δt
Calculating derivatives of 3D cartesian functions
Derivative of each vector component
r’(t)=f’(t)I+g’(t)J+h’(t)K
Unit Tangent Vector
Cartesian Vector divided by the vector magnitude T(t)= r’(t)/|r’(t)|
Calculating indefinite Integrals of 3D Cartesian functions
Integral of each vector component plus a constant
∫r(t)=F(t)i+G(t)j+H(t)k+C
Calculating definite Integrals of 3D Cartesian functions
Integral of each vector component over the same interval
∫r(t)=∫f(t)i+∫g(t)j+∫h(t)k
Constant radius rule
If the radius goes unchanged
r⊙v=0
Net force vector
Object mass (M) times acceleration (a: often given as r” or v’ even in 3d space) equals the net vector force magnitude acting on that object
M*a=|v|
Always in the velocity direction
Trajectory
Function that describes the height and distance of an object with a change in position in 3d cartesian space
Constants when integrating vector valued functions
Always the initial value of the integral
Initial velocity or position
Rule of constant acceleration
When calculating trajectories in a gravitational field, acceleration is always constant (usually 9.81m/s^2) while the initial velocity is almost never zero
Maximum height in trajectories
(sin(a)|Vo|)^2/19.62
Time of flight from gravitational trajectories
(2|Vo|sin(a))/9.81
Range in Gravitational Trajectories
(Sin(2a)|Vo|^2)/9.81
Calculating directional force of the vector given time
F=[Mf”(t),Mg”(t),Mh”(t)]
Mass multiplied by each of the directional acceleration functions, given time
Factors in trajectory launch
Radius is determined by:
Angle of elevation (a) and
Initial velocity (Vo) -barrel velocity
θ angle from x-coordinate, determines position
Can calculate using spherical coordinates
Adjusting for trajectory wind
Given the vector of the wind
Over compensate with your angle measures
Arc length of 3d vectors
∫√(f’(t)^2+g’(t)^2+h’(t)^2)dt
From a to b
Trajectory velocity
√(f’(t)^2+g’(t)^2+h’(t)^2)