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Introduction

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.

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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,

ti = 0

tf  → t

 

In about 80% of the applications, the origin of the motion can be considered at the origin of coordinates; thus, the displacements are xi = 0 xf  → 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

vi  → v0

vf  → v

Here, v0 is called the original velocity
For the acceleration, the average acceleration is the same as the instantaneous acceleration because the acceleration is constant    

 

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Formulas for Uniformed Accelerated Motion

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.

  1. 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.

Motion Application I

  1. 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.

Motion Application III

  1. 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.

Motion Application IV

  1. 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.

Motion Application V

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com