Equations of motion in special case

 In physics, there are several special cases of motion that are described using specific equations of motion. These special cases include:

Uniform Circular Motion: 

  • An object moving in a circular path at a constant speed is said to be in uniform circular motion. 
  • The equation of motion for an object in uniform circular motion can be described using centripetal acceleration, which is defined as the acceleration of an object towards the center of a circle.

a = v^2/r

Where:
a is the centripetal acceleration
v is the velocity of the object
r is the radius of the circular path

Simple Harmonic Motion:

  • Simple Harmonic Motion refers to the periodic motion of an object about its mean position, characterized by a sinusoidal motion. 
  • The equation of motion for an object in simple harmonic motion can be described by the following equation:
  • x = A cos (ωt + φ)


Where:
x is the displacement from the mean position
A is the amplitude of the motion
ω is the angular frequency of the motion
t is time
φ is the phase constant

Projectile Motion:

  • Projectile motion refers to the motion of an object that is projected into the air and subject to the acceleration due to gravity. 
  • The equations of motion for an object in projectile motion can be described by:

x = x0 + v0t cosθ
y = y0 + v0t sinθ - (1/2)gt^2

Where:
x is the horizontal displacement of the object at time t
y is the vertical displacement of the object at time t
x0 is the initial horizontal position of the object
y0 is the initial vertical position of the object
v0 is the initial velocity of the object
θ is the angle at which the object was projected
g is the acceleration due to gravity
t is time

Relative Motion: 

  • Relative motion refers to the motion of an object relative to a reference frame. 
  • The equation of motion for an object in relative motion can be described by considering the motion of the object with respect to the reference frame. 
  • The equations of motion are dependent on the choice of reference frame and the relative velocity between the object and the reference frame.