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Projectile Motion[edit]

The professor is giving a lecture.

"I'm sure you know the three laws of motion. Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it. In a static-velocity measurement-frame of identities, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Velocity is proportional to momentum, depending on the mass of an object. For a constant mass, force equals mass times acceleration, which equals a change in momentum. In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a in which the object goes: F = ma. If the forces add up to 0, the change in momentum is 0, so the acceleration is 0.

You see me launching this ball straight up into the air. You can imagine using a measuring stick to see how high it goes from its initial position and you can use a timing device to see how long it takes for it to reach the top. (If I threw the ball exactly the same way for a few measurements, you should be able to propagate the uncertainty of your measurements in your calculations to state the precision of your measurement of the phenomenon as the magnitude of the result times the square root of the sum of each of the dimensions of each uncertainty in measurement in relation to the final result. e.g. for time x and volume y, uncertainty in flow rate Δr = √[ (∂r/∂xΔx)2 + (∂r/∂y1Δy1)2 + (∂r/∂y2Δy2)2 + (∂r/∂y3Δy3)2 ].)

Velocity is defined as displacement over change in time. It takes about one-half second for it to travel vertically one meter. That means the average velocity is 2 m/s. If you look at the displacement over the one-second period it takes to return to my hand, the scalars +1m and 1m net 0m. So the average velocity of the projectile in motion is 0m/s. If you take the limit of the displacement with respect to time at any point during its motion you get the instantaneous velocity.

Observe this projectile essentially launched perpendicular to the ground with a continuous x-velocity a until it hits the floor at time t1. Note the position, velocity, acceleration, and jerk within the relevant timeframe. Velocity is the first derivative of position with respect to time, acceleration is the second derivative with respect to time, and jerk is the third derivative with respect to time. You can take the derivative of the y-component of the projectile's velocity to find that its acceleration is a constant 9.981A8 m/s2. The derivative of that is 0 jerk. You can also integrate the velocity function to find its relative position graph. I threw the projectile from 1m above the ground, so you can graph the position relative to the floor. Consider these four graphs of a particle traveling upwards over time with varying velocity, acceleation, and jerk.

Chronogrammes loi sinusoidale par partie en vitesse.svg

The force of gravity is proportional to the acceleration by a factor of the mass, which is just 10kg in this case because the projectile is 84g. The force of gravity lowers the vertical component of acceleration from 0, so the velocity becomes negative in the z-direction. It does not change the velocity of the projectile in the x-direction because gravity only operates on the y-axis.

We apply an initial force at an angle b to launch at the velocity a until it reaches a peak at time t and hits the ground at time 2t. The independence of the coordinate axes allows for the constant force of gravity to control the object's velocity vertically, while the horizontal velocity stays the same during its path in the air. The time of impact is given by. So the range is given by. And at half of that time the maximum height of the object is given by. If we measure the angle and inital velocity as, we find. Do you have a question?"

"Yes. Why is the velocity of the object in the x-direction independent of the velocity in the orthogonal dimension?"

"Any vector can be expressed as the sum of a scalar times the coordinate unit vectors. An identical position vector of an object's path from the start to the top of the arc can be extended into the x dimension by changing the initial force launching the object along a varied trajectory with higher speed without deviating from the y component of original vector. This applies not only to position vectors, but any vector, including velocities. So the force of gravity only has an effect in the y component of the net force on the object. That effects its vertical acceleration, enabling the velocity to become negative. This is independent of the net acceleration in the x direction."

Anyway, consider the projectile being launched at an angle into a crevice. The only significant applied force is gravity. Notice that the result is independent of the mass.

General relativity.

For every action there is an equal and opposite reaction. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.


volume ëhilb minim ëhalb milliliter ëhulb barrel
velocity ëhild bubnoff unit ëhald meter per second ëhuld mile per hour
acceleration ëhind bubnoff unit2 ëhand meter per second2 ëhund mile per hour2
mass ëhig Planck mass ëhag kilogram ëhug solar mass
distance ëhib Planck length ëhab meter ëhub parsec
(SSD2) ëhiub/ëheab inch ëhaib/ëhaub rod ëhuib/ëhoab league
time ëhid Planck time unit ëhad hour ëhud calendar year
(SSD2) ëhiud/ëhead sidereal year ëhaid/ëhaud age ëhuid/ëhoad eon
planar angle ëhifw point ëhafw radian ëhufw degree
solid angle ëhafy steradian
frequency ëhadv hertz
aperiodic time ëhižd rutherford ëhažd becquerel ëhužd curie
area ëhimb barn ëhamb hectare ëhumb myriad
(SSD2) ëhiumb/ëheamb sq. inch ëhaimb/ëhaumb sq. mile ëhuimb/ëhoamb township
atomic mass ëhaltʰ dalton (u)
energy ëhibv Planck energy ëhabv joule (J) ëhubv quad
force ëhizd Planck force ëhazd newton (N) ëhuzd ton-force
pressure/stress ëhizb bar ëhazb pascal (Pa) ëhuzb standard atmosphere
power ëhigr ton of refrigeration ëhagr watt (W) ëhugr horsepower
temperature ëhibb Planck temperature ëhabb kelvin (k) ëhubb degree Réaumur
luminous intensity ëhabz candela (cd) ëhubz candlepower
luminous flux ëhabl lumen
luminous exitance ëhabr lux ëhubr foot-candle
luminance ëhibž stilb ëhabž candela ëhubž bril
airglow ëhabd rayleigh
amount ëhagg mole ëhugg International Unit
density ëhadg gram/mole
catalytic activity ëhagd katal ëhugd enzyme unit
information ëhigž dit ëhagž nat ëhugž bit
probability ëhadn probit
data transmission speed ëhažb baud
loudness ëhidr sone ëhadr decibel ëhudr phon
acoustic absorption ëhagdh sabin
osmotic pressure ëhabdh osmol
fineness of precious metal ëhadl karat
permeability of a porous material ëhagb darcy
quantized magnetic moment ëhadb magneton (J/T)
currency ëhidm Yuan (¥) ëhadm Euro (€) ëhudm US dollar ($)
(SSD2) ëhiudm/ëheadm Mexican peso ëhaidm/ëhaudm Australian dollar ëhuidm/ëhoadm South Korean won


Friction on an Inclined Plane[edit]

Tension on an elevator and three-way tug-of-war[edit]

Centripetal acceleration on carnival rides and Orbital gravitation[edit]

Periodic motion and frequency

Kinetic, Potential, Spring energy and Power (LOL diagrams on simple machines)[edit]

Collisions Torque and Center of Mass (Ball hits rod)[edit]

Gears and Moment of Inertia for Perfect Rolling[edit]

Harmonic Oscillators and Multiple Pendulums[edit]

Simple harmonic oscillator, amplitude, period, phase, pendulum, simple harmonic motion,

Sound and Music[edit]

Speed, relative speed, Mach, Decibel, Ultrasound, Doppler Effect, Interference, Standing waves on strings, open tubes, closed tubes, beat frequency.

Buoyant force and Fluid dynamics[edit]

Specific Gravity, Pascal's Principle, Archimedes Principle, Buoyant Force, Equation of Continuity, Bernoulli's equation, Viscosity, Poiseuille Flow, Reynold's Number, Venturi Effect, Pitot Tube, Surface Tension and Adhesion.


Gases, Maxwell Boltzmann Distribution, Specific Heat, Latent Heat of Fusion and Vaporization, Thermal Conduction, Convection, Radiation, Quasistatic/reversible processes, PV diagram, Isothermic Process, Carnot Engine, Entropy, Efficiency, Maxwell's Demon

Special Relativity[edit]

Einstein Field Equation symmetry. Lorentz Transformation.

Light, Optics, Electricity and Cosmology[edit]

Lagrangian Mechanics[edit]