KINEMATICS: Matching Equations to Appropriate Circumstances ( Part 1 )

Of all the topics that cause confusion among students new to physics, kinematics is no exception to the rule. Briefly speaking, kinematics can be described as the “ architecture of motion. “ Various types of forces ( F ) and energy ( E ) can give rise to motion observed within a system, whether that motion be linear or curvilinear, or whether the object moves with a constant velocity ( v ), acceleration ( a ), or some combination of these aforementioned values. Additionally, the degree to which an object is displaced ( s ) horizontally along the x-axis, vertically along the y-axis, or diagonally within a coordinate plane will vary according to circumstance. These are some of the concerns we address via the usage of kinematic equations; these equations model the interaction of various types of motion with very little ( if any ) concern for what type of force or energy gives rise to such motion. Furthermore, these occurrences occur in relation to the passage of time ( t ), so time itself may ( or may not ) be a factor used when solving problems. Equations that model the architecture of motion can and oftentimes include some combination of the following equations:

v = v_o + at

v_av = ½ ( v_o + v )

s = ½ ( v_o + v )( t )

s = v_ot + ½ at²

v² = v_o² + 2as

The small ( o ) or “ knot “ subscript is used to denote original value ( Ex. v_o = Initial velocity ).

Note: Although ( s ) is generally used to describe distance or displacement in kinematics, other variables such as ( d ) may be encountered. As far as horizontal and/or vertical displacements along the x-axis or y-axis are concerned, the following are examples of equations that are always encountered, yet they are encountered in accordance to the circumstance at hand:

Horizontal Motion

x = v_oxt + ½ at²

v_x = v_ox + at

v²_x = v²_ox + 2ax

Vertical Motion

y = v_oyt + ½ at²

v_y = v_oy + at

v²_y = v²_oy + 2ay

In their present form, the plus ( + ) sign suffices very well; however, what about circumstances where an object may be decelerating??? How will the acceleration terms differ when some force acts against the direction through which an object is being displaced??? In these circumstances, we simply insert a negative value into the equation assuming that the direction of motion has been assigned a positive value along with a positive velocity value associated with value designation:

Ex. v_x = v_ox + at

If this equation is used to describe the motion of an object that is slowing down, the corresponding acceleration value will be negative:

v_x = v_ox + ( – a )( t )

v_x = v_ox – at

When ( t = 0 ), the ( v_x ) term describes the instantaneous ( and initial ) velocity ( v_ox ) being traveled at that instant in time. As time transpires, the ( – at ) term will grow larger and more negative, because time is always assigned a positive value. As a consequence, the value of ( v_x ) will lessen as time transpires. If enough time transpires, the object’s velocity will be reduced to zero. If more time passes than is needed to decrease an object’s velocity to zero, the ( v_x ) term will become negative, indicating that a reversal of motion has occurred. This scenario describes how speed and velocity differ; speed is a designation of the magnitude to which an object’s position is changing over time, whereas velocity assigns both a magnitude and direction to such motion. Let’s now look an an example where acceleration is due to the gravitational constant of acceleration ( g ) close to the earth’s surface:

Ex. y = v_oyt + ½ at²

If we designate the +y-axis to describe a positive displacement of an object in the vertical direction, we must account for the fact that acceleration due to gravity is in direct opposition to this motion; thus, it is appropriate to substitute the ( a ) term with a ( – g ) term:

y = v_oyt + ½ at²

y = v_oyt + ½ ( – g )( t² )

y = v_oyt – ½ gt²

Once the initial positive and negative signs are attributed to each term within any kinematic equation, it is crucial to stick with these designations throughout the duration of problem solving!!! It is typical to use the +y-direction to denote upward motion, and objects moving horizontally towards the right are typically designated as moving in a positive direction as well.

Before matching some of the aforementioned equations to circumstances in which they may be relevant, let’s assure that our understanding of acceleration is as unambiguous as it may be. An acceleration is a measure of how much an object’s velocity changes over time:

a = (  Δv / Δt )

Since velocity is expressed in terms of meters per second, a more general form of this equation is as follows:

a = [ ( m / s ) / s ]

Thus, acceleration describes an object’s changing velocity in meters per second over every second it travels:

Q. If an object experiences a change in motion equivalent to 2 meters per second every second, what is the magnitude of its acceleration?

A: a = [ ( 2m / s ) / s ]

Q: At what velocity will the object in question move after accelerating for 2 seconds?

A: v = at

v = [ ( 2m / s ) / s ]( 2s )

v = 4 m/s

Fortunately, the rules of algebra allow us to change the equation from its current form via multiplication of the numerator by the inverse of the term in the denominator:

a = [ ( m / s ) / s ]

a = ( m / s )( 1 / s )

a = ( m / s² )

Now that a basic foundation for the kinematic equations has been established, we are ready to see how to get a good idea of which ones to choose when addressing any of several circumstances.