THE TRUTH in 3D Velocity and Acceleration Equations

Morpheus, after running the “Instant Kesten Program,” approaches his student Neo with two pills, a red one and a blue one.

He states that if he swallows the blue pill he will forget what has happened and return to his life, but if Neo swallows the red pill then THE TRUTH will split into multiple dimensions.

Neo swallows the red pill and suddenly the world becomes a bit more complicated, THE TRUTH breaks apart into

vx=vox
vy=voy-gt

x-x0=voxt

y-y0=voyt-1/2gt2.

The red pill makes him understand that horizontal movement has no acceleration due to gravity (and generally in simpler problems), but that vertical motion is affected by gravity.

Neo realizes that two dimensions, x and y, can also be combined into the formula

y – y0=(x-x0)tan(θ) – [g(x-x0)2]/[2v02cos2(θ)] when the acceleration present is gravity.

Morpheus tells him as well that multidimensional physics necessitates proper labeling of variables like time in order to be accurate.

Neo says “whoa.”

Basketball “Hang Time” Physics

Michael Jordan Flies Through the Air (statue) from Esparta on Flickr

Michael Jordan Flies Through the Air (statue) from Esparta on Flickr

The stadium is packed with fans as Michael Jordan flies through the air on his way to the rim. Gravity seems suspended in this “hang time” phenomenon.

The hot dog dealer casually explains that by using THE TRUTH, and possibly a know/don’t know table, it can easily be calculated that it is true that Jordan spends 2/3 of the time actually in the air and with a vertical leap of over forty inches this can be a long time.

The hot dog dealer continues to explain the physics of the game by showing how vector analysis can track the motion of basketball stars.

A vector has direction and magnitude and can be added or subtracted using normal additive rules.

Thus he explains that the offense can jump both forward and up, but if the defense has the same vertical component they will both land at the same time since horizontal and vertical motion components are independent.

The dealer then continues on his way and warns not to use tangent for the x or y components

Instead, multiply the vector by sin(θ) to find the y component (magnitude in the y direction) and multiply the vector by cos(θ) for the  x component (magnitude in the x direction.

Also, by convention you should measure θ from the positive x-axis (this is physics convention anyway).

Michael Jordan top 40 moments

Falling Velocity – Breaking a Container

Late one night a group of physics students dressed all in black and equipped with night vision sneak into Kesten’s office and steal a metal container labeled “Chug’s Secrets.”

They learned the a2 = (a+b)(a-b)+b2 rule for squaring numbers and thought that they might learn amazing tricks for mastering physics by seeing how Chug worked his magic.

By using THE TRUTH these students calculated that they could break the metal container by dropping it from the top of (four story) Casa Italiana, but unfortunately they didn’t look at the situation carefully and in their calculations placed y0 at ground level and had a positive acceleration for gravity.

The final velocity was not sufficient and the container did not break, luckily a former student of Kesten heard the clang of the metal and helped the students understand their computational errors and also advising them to go to 11 story Swig.

At Swig the group had a baseball player throw the container as fast as he could down at the ground, since a higher v0 in the right direction will produce a higher velocity.

The container broke on impact!

Unfortunately the final velocity was so strong that the police heard the noise and took the container to Area 51, since then “Chug’s Secrets” have still not been uncovered.

One Dimensional Motion Formulas

v = u + at

v2 = u2 + 2as

s = ut + 1/2 at2

average velocity = (v + u)/2


v: velocity (this is the velocity u combined with increases or decreases due to acceleration)

u: velocity (in this case before any acceleration)

a: accelation

t: time

s: distance


Derivatives and Differentials

Differentials and Derivatives

A. Letters u and v denote independent variables or functions of an independent variable; letters a and n denote constants.

B. To obtain a derivative, divide both members of the given formula for the differential by du or by dx.

derivatives_1

Differentiation of Integrals:

If f is continuous, then

derivatives_2

Chain Rule:

If y = f(u) and u = g(x), then

derivatives_3

Greek Alphabet

Greek letters surface all the time in physics, here are the 24 letters in capital and lower case, along with their names.


Alpha α Α
Beta β Β
Gamma γ Γ
Delta δ Δ
Epsilon ε Ε
Zeta ζ Ζ
Eta η Η
Theta θ Θ
Iota ι Ι
Kappa κ Κ
Lambda λ Λ
Mu μ Μ
Nu ν Ν
Xi ξ Ξ
Omicron ο Ο
Pi π Π
Rho ρ Ρ
Sigma σ Σ
Tau τ Τ
Upsilon υ Υ
Phi φ Φ
Chi χ Χ
Psi ψ Ψ
Omega ω Ω