Newtons Second Law F=ma

As the village blacksmith walks along one day he sees a group of monks circling around an apple tree and chanting something about the truth being expanded.

When he stops and listens he learns that a man named Newton became inspired by falling of apples.

The monks explain that Newton then came up with the equation ΣF = ma (where F and a are both vectors) which means that the sum of the forces in a direction is equal to the mass, not to be confused with weight, multiplied by the acceleration, one example being gravity, in a direction.

The F and a are vectors, indicated by bold.

The monks are rejoicing since they had only previously been able to understand the motion of objects.

Now they can understand interaction of objects with mass through the use of vector components, the old testament of the truth, and their newfound equation.

Circular Motion Centripetal Acceleration Arc Length

One day a little kid is playing with a yo-yo and swinging it around in a circle for his own amusement.

Suddenly a crazed physicist walks up and decides to explain the physics of the circular motion.

First he asks the kid to spin the yo-yo around without any acceleration and the kids tries, but this turns out to be a trick question since there always is an acceleration towards the center (centripetal or center seeking force) because the velocity changes directions.

The physicist next shows how components can be used for measuring velocity vectors.

He then explains that describing certain aspects of circular motion with rectangular coordinates is pure silliness and that sometimes polar coordinates with an angle and a radius are much simpler.

He notes however that unlike in navigation, the angle is measured from the positive x-axis in physic.

Finally the physicist gives a nifty trick (s=rφ) for finding the arc length of pizza, or any circle, if one knows the radius and angle phi, but speaking of pizza he thinks about how good some would be and walks off.

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


Relative Motion & Reference Points

When something is observed it is always observed from some vantage point.

If two people are running next to each other it will not seem as if the other is moving, since their relative motion is the same.

Since the motion is really the same from any vantage point the reference point used to analyze physics can be chosen to be anywhere.

However, some reference points are more useful than others (depending on the situation).

For instance, with gravity- the lowest point in a certain situation could be ten meters above ground if an object is sitting on top of a roof, thirty two meters below ground if there is a hole, or at ground level.

Setting a convenient reference point will often make the manipulation of equations simpler though.

For instance, setting the reference point on the ground=0 level can make an integral easier because the bound of zero may cancel out terms.

Having zero at ground level may not always be convenient though, and always using zero to cancel out terms can be deceiving since it will not always work that way.

Car velocity physics – physics velocity

A physics professor drives along in his camry on the highway and covers about 88 feet per second or a velocity of 100 km/h when he observes a mitsubishi GT 3000 that races out onto the freeway.

This vehicle has an initial velocity of 100 km/h and smoothly accelerates, with the increase in velocity being constant over time, up to 250 km/h which is about 220 ft/s.

A police helicopter spots the GT 3000 and calls in squad car to lay a spike strip. The GT 3000 slams on the breaks and leaves skid marks for 160 feet. The police officer who finds the driver calmy whips out the stone tablet of THE TRUTH (v=v0 + at and x-x0 = v0 + 1/2 at2) and using this and his knowledge of the GT 3000’s breaking capabilities calculates the initial velocity and cuffs the driver for reckless driving.

The physics professor continues at a constant velocity on the highway, passes the arrest, and hears the driver cursing physics, but he knows the power of THE TRUTH.

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

Dimensional Analysis in Physics – Checking Answers

Dimensional analysis

Dimensional analysis allows a quick check of your answer by seeing if the units are correct.

An example would be if you are finding a distance by multiplying a speed by a time.  The units of distance are meters and the dimensional analysis of the product is [velocity*time] = [(m/s)(s)] = [m]

Google can do this automatically (with SI units) for an even faster check.