I.   Introduction

A motion occurs when position changes in time.  Motions are initiated, modified,  or terminated by forces.  A force  is that which causes or modifies a motion.  A general motion is a combination of  a  translational and  a rotational motions.
 
In a translational motion, every point of the body moves in a straight line and every point experiences the same linear displacement (change in position) during a given time interval.  In a translational motion, the body moves along fixed, straight line.
 

 Example:  A person moving along a straight line. An object sliding down  an inclined plane.

 Example:  Motion of wheels, of hands on a clock,  of spinning electrons,    of planets around the sun,  of galaxies,etc.

What is acceleration?  Acceleration is the rate of change of  the velocity;
v or v(t) is the x-component of the velocity. And a is the x-component of the acceleration.
Given 

Note:  v can either be less than 0 (v < 0) or greater than 0 (v > 0) and .

   and .

Two basic equations for linear uniformly vaired (LUV) motions are:

The latter equation is commonly known as the Golden Rule.

These two equations can be used to derive all other equations for a linear uniformly varied motion--as shown below !

II. Linear Uniformly Varied Motion

A linear uniformly varied motion is linear motion with  .

Equations of motion:

If v0 = 0 , then 

If x 0 = 0 , then 

If x 0= 0 and v0= 0 , then 

If and are in the same orientation, the motion is uniformly accelerated.
If and are in opposite orientation, the motion is uniformly decelerated.

If the axis is in the direction of , then and 

III. Linear Uniform Motion

 A linear uniform motion is (a linear) motion with   In order for a body to  have zero acceleration, the body must either be at rest or have a constant   velocity.
 
If (zero is a constant)

Equations of Motion: 

1.) An object is dropped at a height of 2m above the ground.  Find the   time the object takes to reach the ground.  Find the speed at ground level.

The speed at ground level is 6.26

2.) An object is thrown up vertically  with an initial speed of 10m/s .  Find the maximum height it reaches.  Find the time to reach that height.

Equation:  here, a < 0  so  a = - g  if the direction of the x-axis: upwards.

At point B, v = 0 . 

Notice: 

Another method to find h: 

IV. Circular Motion

 Circular motion occurs when a particle moves along the circumference of a fixed circle (of fixed radius r).

V. Uniformly Varied Circular Motion

 A uniformly varied circular motion  is a circular motion with a constant length (magnitude) for the tangential acceleration, i.e.,

Note:    is the magnitude or length of the vector.  It is constant.
  is a vector, it is not a constant because its direction is  changing.

In other words, in a uniformly varied circular motion, the tangential acceleration is constant in magnitude, but not direction. Please distinguish the total acceleration from the tangential and centripetal accelerations. (See the diagram above).
 
 

Equations of Motion: (for a uniformly varied circular motion)

For r constant, we have  .
 
Golden Rule:

The key point about the equations for circular uniformly varied motions is that (a) they are similar to the ones for a linear uniformly varied motion (if you change x to s and  a to  and (b) in changing from the linear variables (s and v) to the angular variables one only needs to know where  is always in radians (rds).
 
 

VI. Uniform Circular Motion
 
 A uniform circular motion is circular motion with 

Note:  s = arc length, t = time, and .
 

Please note well that the equations describing a motion with constant magnitude for the tangential acceleration also apply to a uniform circular motion! zero, after all, is a constant. So, please derive the needed equations from the ones given above for a uniformly varied circular motion.