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Projectile Motion[edit | edit source]

The professor is giving a lecture: "We will now learn about objects and systems which have properties like mass and charge. Systems may have object interactions. We analyze objects mechanically via relationships between their properties and we analyze systems temporally via conserved quantities. Fields in space can explain interactions. Object-object interactions can be described by one of the four forces and system-system interactions can result in internal changes constrained by conservation laws. Later I'll discuss how waves can transfer energy and momentum from one place to another without permanent transfer of mass and how probability can be used to describe quantum phenomena.

I'm sure you know the three laws of motion. The first law of motion asserts that 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 spacetime-coordinate-system, an object either remains at rest or continues to be displaced at a constant rate of travel, unless acted upon by a force. This is illustrated in Fig. 1 where a ball rolling on a frictionless surface has the same velocity after four seconds of time t. There are no forces providing it with acceleration a. If one blew a fan at it in the direction of its travel, it would accelerate and the distance over time would increase. If one blew the fan in the opposing direction it would decelerate. If it were not resting on the ground it would accelerate downwards at 9.81 m/s² due to the force of gravity.

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.

In Fig. 1 the green observer's reference frame is stationary and sees the apple fall at the same velocity as the elevator, whereas the blue observer's reference frame has the same velocity as the elevator and sees the

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.

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 9.981 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.

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.

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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.

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Friction on an Inclined Plane[edit | edit source]

Rolling Racers - Moment of inertia.gif
Box on an Inclined Plane.png
A laboratory manual in physics, to accompany Black and Davis' "Practical physics for secondary schools," (1913) (14764785362).jpg

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

Centripetal acceleration on carnival rides and Orbital gravitation[edit | edit source]

Mass Spring System Resonance.gif

Periodic motion and frequency

Static friction angle.jpg
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Moon orbit.jpg
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Earth tilt sample.jpg

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


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

Torque animation.gif
Elastic hit.png

Gears and Moment of Inertia for Perfect Rolling[edit | edit source]

Harmonic Oscillators and Multiple Pendulums[edit | edit source]

Mass Spring System Undamped case Simple harmonic motion.gif

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

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Standing wave in a rope.png
Standing waves on a string.gif
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Interference of two waves.svg
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Sound and Music[edit | edit source]

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 | edit source]

Surface Tension Diagram Buoyancy.svg

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.

Thermodynamics[edit | edit source]

Carno engine.PNG

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, semiconductors

Particle Physics[edit | edit source]

strong, weak forces, beta decay, Feynmann diagrams

Special Relativity[edit | edit source]

Einstein Field Equation symmetry. Lorentz Transformation.

Relativity-of-simultaneity white.svg

Light, Optics, Electricity and Cosmology[edit | edit source]

Two Slits Paradox.png

luminosity, electromagnetic radiation, circuit diagrams, motors and generators, EM fields

Lagrangian Mechanics[edit | edit source]