Motion of a ball falling

This video represents the motion of a ball falling to the ground.  The motion is represented by -mg where m is the mass of the ball and g represents gravity (9.8m/(s^2)).  As the ball is dropped from reset, it slowly starts to fall at first, then picks up speed faster and faster until it hits the ground.  

This can also be represented by potential and kinetic energy.  Conservation of energy says that energy cannot be created or destroyed.  This means that As potential energy decreases, kinetic energy increases.  At the top before the ball is dropped,  this is the point of maximum potential energy.  The ball has no velocity or acceleration, so the kinetic energy at this point is 0.  As the ball is dropped, the velocity and acceleration of the ball increases causing it to have kinetic energy so the potential energy decreases.  

Projectile Motion of a baseball with wind resistance

This animation shows the projectile motion of a baseball with wind resistance.  The equation for wind resistance is represented by Fwind= -0.5CρAv^2.  C is the coefficient of performance, ρ represents the density of air, A represents the frontal area of the baseball and v represents the velocity of the wind.  Other forces acting on the baseball is shown by the equation Fgravity= -mg.  

The baseball has an initial velocity = {10m/s, 0, 10m/s}.  This means that the baseball has an initial velocity in both the positive x and z direction, but not in the y direction.  The forces opposing the baseball are wind and gravity which can be represented by the vector d = {Fwind, 0, Fgravity}.  These forces are both in the negative x and z direction, but not in the y direction.  This is not practical since wind does not only blow in one direction.  We can however set the opposition y force to 0 to better show how the baseball can be effected by the force of wind.  The wind opposes the velocity of the baseball, causing it to have a negative acceleration and changing direction of the baseball as you can see in the animation above.  

Simple Harmonic Motion

A good example of simple harmonic oscillation is in a spring system.  You can see an example of this in the video at the bottom of the blog.  These systems are represented by the formula: a+2βv ̇+w0^2x=Acos(wt). 2βv represents the dampening force, Acos(wt) represents the driving force and w0^2x represents the spring force.  The w0^2 shows the natural frequency (the frequency the oscillator will continuously oscillate at without any external force) of the oscillator while x represents the displacement.  If 2βv = 0 and Acos(wt) = 0, the motion of the spring is represented by:

This graph has no change in displacement because no outside force is acting on the system.  If 2βv does not equal 0, but Acos(wt) = 0.  The oscillator will start to get slower and slower until it comes to a complete stop.  This is because the dampening resists the motion of the spring.  This is represented by:

As you can see, the spring oscillates for a couple of seconds then comes to a complete stop.  This is because there is no driving force to help keep the system in motion.  If both 2βv and Acos(wt) do not equal 0, then the displacement of the spring will be off in the beginning but eventually even out and reach simple harmonic motion.  This is shown by the following graph:

(video obtained by physics animation channel on youtube)