

In this section, the study of one dimensional uniform
accelerated motion is presented. This motion should be considered an application
of accelerated motion. Here, even when same of the mathematical derivations may
work just the same for motions with more than one dimension, only one
dimensional motion is presented. In addition, the word uniform means is used to
indicate constant. Thus, uniform accelerated motion means motion with constant
acceleration.

Change in Notation

To emphasize the notion that this motion is not a general
motion, the following simplifications and change in notation are considered,
The time is
measured with a stop wash rather than with a clock, the implication is
that the time is always measured from zero; thus, 
t_{i}
= 0 
t_{f}
→ t 

In about 80% of the applications, the
origin of the motion can be considered at the origin of coordinates;
thus, the displacements are 
x_{i} = 0 
x_{f} →
x 
Notice that no vector notation is required because of
the single dimension being studied 
In the case of the
velocity, to assume that the initial velocity is zero restricts
the variety of motion excessively; however the following change in
notation is introduced 
v_{i}
→
v_{0} 
v_{f}
→
v 
Here, v_{0}
is called the original velocity 
For the
acceleration, the average acceleration is the same as the
instantaneous acceleration because the acceleration is constant 



Based on the previous notation, the following formulas can be obtained:
Definition of average velocity 

=> 

(U_A_M 1) 
Between the starting of the motion and the motion at
the present instant of time, the average velocity can also be defined
based on the characteristic formula for the average between two
quantities. 

(U_A_M 2) 
Notice that this formula is only valid when the
acceleration is constant because, in this case, the numerical weight of the
two velocities is the same. In
another page, there is an application where this
formula gives the wrong answer to the average velocity because the
mobile
acceleration is not constant. 
(U_A_M 2) 
From the definition of instantaneous acceleration,
which in this case is the same as the average velocity, a formula for
the velocity can be derived 

=> 

(U_A_M 3) 
The displacement at any instant of time can be
calculated for an object starting from (U_A_M 1)
and substituting (U_A_M 3) 

=> 

(U_A_M
4) 
Also, a formula that is not explicitly time dependent
can be obtained as follow: 
Starting from (U_A_M 3)
and multiplying both sides by the average velocity 
=> 

(U_A_M
5) 
The above formulas can be used for solving many problems
involving uniformed accelerated motion in one dimension. The difficulty of
problems and or exercises can be categorized accordingly with the number of
formulas that need to be used in order to obtain the requested answer. From the
point of view of who is solving the exercise, you do not have real control on
which formula should be used for solving the exercise all the control is on the
person that designed the problem or exercise. Your understanding of the subject
will be the tool applied in discovering the formulas that the designer of the
problem expect you to use at the moment of solving the problem.

The simplest exercises are those that request a direct
substitution of the known values into one of the formulas in order to obtain
the requested answer. Inside this group, the different difficulties are
associated with the particular formula that you are required to use in order
to solve the exercise. In this case, to use formulas such as (U_A_M 1), (U_A_M
2), or (U_A_M 3) are considered the
simples exercise. Consequently, the use of formulas (U_A_M
4) and (U_A_M 5) is considered harder.
Additional difficulties may be present if change of units are required prior
or after solving the exercise. In this case, usually it is simpler when the
exercise just requires changing the units at the end rather than at the
beginning.

The next level of difficulty can be identified when the
problem requires that the unknown variable to be solved from one of the
formulas above in order to obtain the solution of the problem. To simplified
the solution of this exercises, you may choose to first substitute the
numerical values of the known
quantities in the formula in order to obtain a numerical equation in the
unknown rather than a literal equation. However, solving the exercise using
literal equations rather than numerical equations may be more convenient
when trying to retrace or check the obtained solution. In this notes, the
problems will be solved following the solution of a literal equation except
for the cases in which such approach extend the achievement of solution
unnecessarily. Just as in the group before, the most difficult problems in
this group are those involving formulas (U_A_M 4)
and (U_A_M 5). In particular, to use (U_A_M
4) for solving for the time may be difficult because it involves to
solve for a quadratic equation.

More complicated problems are those where more than one
equation is necessary for obtaining the solution of the problem. In this
type of problem, in order to obtain the solution, it is necessary to
calculate an intermediate quantity that it is also unknown. Therefore, at
least two formulas will be necessary for solving the problem. The formulas
should always be selected trying to minimize the number of calculations
involved. An example of this method, it will require to select the equation
for obtaining the solution expecting to have no more than two unknowns, one
of these unknowns should be the requested solution and the other one should
be an unknown for which its value can be easily obtain from another formula
for which all the other quantities are known. In most cases, one of the
unknown can be obtained from one of the simpler formulas (U_A_M 1), (U_A_M
2), or (U_A_M 3) and the one that solve
the problem using one of the five formulas associated with uniformed
accelerated motion. As in the previous cases, the simpler the formula the
better for solving the problem.

Additional difficulties my be encounter in problems that
require solving for different mobiles in order to obtain the necessary
solution. For motions that include more than a single mobile, the level of
difficulty can also be classified accordingly with the previous three groups
but with a greater level of difficulty involved.
