In summary, the article discusses the use of completing the square technique to solve problems involving the Newtonian equations of motion. It provides an example of applying this technique and also discusses the importance of considering the units in the final result.
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Introduction
In any school Physics course, the Newtonian equations of motion are very much a ‘stock’ item. Students learn the equations and are given a variety of problems that provide practice in determining which equation(s) to use to solve any particular problem. What is perhaps a little surprising is that in general no one applies the well-established mathematical technique of completing the square given the standard equation: ##s(t)=ut+\frac{at^2}{2}##. In this article, we will do just that and then apply the derived equations to solve some typical problems which arise.
Completing the Square
Given: ##s(t)=ut+½at^2## write as $$ s(t) = \frac{a}{2}\left(t^2 + \frac{2ut}{a}\right)= \frac{a}{2}\left[\left(t + \frac{u}{a}\right)^2-\frac{u^2}{a^2}\right]$$ In a typical projectile problem, we might define the positive direction as upwards in which case the acceleration parameter g will be negative and the equation may be written as: $$s(t)=-\frac{g}{2}\left[\left(t –...

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Hello, in the calculation of the two proposed examples, the order of the presentation of the development of the calculation of points 2 and 3 or (b and c) are reversed, also the result of the speed is offered in meters and not in meters over seconds in the first example.
 
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Richard R Richard said:
Hello, in the calculation of the two proposed examples, the order of the presentation of the development of the calculation of points 2 and 3 or (b and c) are reversed, also the result of the speed is offered in meters and not in meters over seconds in the first example.
Thanks for the correction(s). Have amended accordingly.
 
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1. What are the equations of motion?

The equations of motion are a set of mathematical equations that describe the motion of an object in terms of its position, velocity, and acceleration. These equations are based on Newton's laws of motion and can be used to predict the future motion of an object.

2. What is the difference between the equations of motion and the laws of motion?

The laws of motion, as stated by Newton, are general principles that describe the behavior of objects in motion. The equations of motion are specific mathematical equations that can be derived from these laws to solve for the motion of a particular object.

3. How many equations of motion are there?

There are three equations of motion, also known as the kinematic equations. These are the equations for displacement, velocity, and acceleration, and they can be used to solve for any of these variables given the other two.

4. What are the variables in the equations of motion?

The variables in the equations of motion are displacement (x), velocity (v), acceleration (a), time (t), and initial velocity (u) or final velocity (v0 or vf). These variables can be either scalar or vector quantities.

5. How are the equations of motion used in real-life applications?

The equations of motion are used in a wide range of real-life applications, including engineering, physics, and sports. They can be used to design and analyze structures, predict the trajectory of projectiles, and optimize the performance of athletes. They are also used in the development of technologies such as rockets, airplanes, and cars.

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