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.

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

Sound Level Terminology

  • Absorption Coefficient- Denoted by alpha (units of inverse meters) and “a” (dB/m). This constant measures the sound-absorbing ability of a material. The values go from about 0.01 for marble slate to almost 1.0 for the long absorbing wedges that are used in some anechoic sound chambers.
  • Anechoic Sound Chamber- room in which the walls, ceiling, and floor all are covered with sound absorbing materials shaped to maximize sound absorption. Echoes effectively do not exist in such a chamber
  • Decibel- logarithmic unit used to gage sound level. Subtracting three decibels translates to reducing the intensity by 50%. Human ears, however, perceive sound that is 10 times less intense as being half as “loud”. In other words, a difference of 10 dB will seem to be twice as loud or quiet.
  • Hertz- unit of frequency (inverse seconds). AKA cycles/second.
  • Infrasonic- sound lower than 20 hertz
  • Medium- the material that something travels through
  • Octave- difference in pitch equal to a doubling of frequency
  • Threshold of pain- 120 dB- the level that goes from discomfort to pain and hearing loss
  • Tone- a definite pitch
  • Ultrasonic- frequencies about 20,000 Hz (20 kHz)

Decibel Level Sound
10 Light Whisper
20 Soft Conversation
30 Normal Conversation
40 Light Traffic
50 Loud Conversation
60 Busy Office
70 Traffic, train
80 Subway
90 Heavy Traffic, Thunder
100 Jet Plane takeoff
120 Pain Threshold

Golf Ball Physics

Golf Balls Have dimples on their surfaces to minimize drag (a force that dissipates when an object moves through a fluid).

A smooth ball causes the air to flow in such a way that the air “sticks” to the ball longer. The dimples also act to create backspin- which makes the air pressure on the top of the ball decrease- giving it lift (somewhat like the situation with an airplane’s wings).

A smooth ball that ravels 65 meters would travel something like 275 meters with dimples when hit with the same force.

Golf balls have 300-500 dimples that can be 0.25 mm deep.