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Chapter 6: Problem 71

In Exercises \(71-76,\) reduce each fraction to simplest form. Each is from the indicated area of application. $$\frac{v^{2}-v_{0}^{2}}{v t-v_{0} t} \quad \text { (rectilinear motion) }$$

Short Answer

Expert verified
The simplified form is \( \frac{v + v_0}{t} \).

Step by step solution

01

Factor the Numerator

Identify the expression in the numerator: \( v^2 - v_0^2 \). This is a difference of squares, which can be factored into \((v - v_0)(v + v_0)\).
02

Factor the Denominator

Identify the expression in the denominator: \( v t - v_0 t \). Notice that \( t \) is a common factor. Factor \( t \) out of the expression to get \( t(v - v_0) \).
03

Simplify the Fraction

Write the fraction with the factored forms: \( \frac{(v - v_0)(v + v_0)}{t(v - v_0)} \). Identify the common factor \( (v - v_0) \) in both the numerator and the denominator.
04

Cancel Common Factors

Cancel the common factor \( (v - v_0) \) from the numerator and the denominator. The resulting expression is \( \frac{v + v_0}{t} \).
05

State the Simplified Form

The fraction is now in its simplest form: \( \frac{v + v_0}{t} \). This is the reduced form of the original fraction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
In algebra, factoring is a method used to break down an expression into simpler components known as factors. This process can simplify expressions and solve equations more easily.
  • Factoring reveals a product of polynomials or numbers that, when multiplied together, give the original expression.
  • Applied to expressions like quadratics, it can simplify calculations and aid in graph plotting or understanding equations more fully.
In the given problem, the numerator, \(v^2 - v_0^2\), has been factored using a special algebraic identity. This process made it possible to simplify the entire fraction by revealing common factors. The key is identifying which terms can be split into simpler multiplicative components. It effectively sets the stage for further manipulation of the expression by isolating like terms.
Difference of Squares
A difference of squares is a specific algebraic form characterized by the expression \(a^2 - b^2\). It can be factored into two binomials: \((a - b)(a + b)\).
  • This technique is named such because it involves subtracting one square term from another.
  • The pattern quickly simplifies complex quadratics or reduces expressions for further calculations.
In our original exercise, recognizing the difference of squares was crucial for factoring the numerator, \(v^2 - v_0^2\). By rearranging it into \((v - v_0)(v + v_0)\), the algebraic simplification was made straightforward. This form is common and useful when solving polynomial equations or simplifying expressions across many areas of mathematics.
Rectilinear Motion
Rectilinear motion describes movement along a straight line. It's one of the foundational concepts in physics when discussing velocity, time, and distance.
  • In problems involving rectilinear motion, equations often describe how one variable depends on another, like velocity and time.
  • Simplifying these equations often aids in deriving meaningful physical insights, such as constant speed or acceleration.
The fraction in the exercise stems from rectilinear motion contexts. It represents a relationship of squared velocities and commonly used time factors. Factored and reduced, it can depict simplified versions of physics problems describing motion, making calculations and interpretations more straightforward.
Fractions
Fractions represent the division of one quantity by another. Simplifying fractions is a key concept that involves reducing a fraction to its most straightforward or smallest form.
  • This process often hinges on identifying common factors in the numerator and denominator and canceling them out.
  • Simplified fractions are not just easier to read; they often improve computations and reveal more about relationships between the quantities.
In this example, the expression \(\frac{v^2 - v_0^2}{v t - v_0 t}\) represents a type of fraction. By factoring and identifying commonalities, the fraction is reduced to \(\frac{v + v_0}{t}\). This intuitive form is easier for problem-solving and calculation, offering clearer insight into the involved variables.

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Most popular questions from this chapter

In Exercises \(71-76,\) reduce each fraction to simplest form. Each is from the indicated area of application. $$\frac{E^{2} R^{2}-E^{2} r^{2}}{\left(R^{2}+2 R r+r^{2}\right)^{2}} \quad \text { (electric power) }$$

Factor the given expressions completely. $$2 z^{2}+13 z-5$$

Factor the given expressions completely. $$8 x^{2}-24 x+18$$

In Exercises 69 and \(70,\) answer the given questions. $$\text { If } 3-x < 0, \text { is } \frac{x^{2}-9}{3-x} > 0 ?$$

Solve the given problems. Is it true that \((1-2 x)^{2}=(2 x-1)^{2} ?\)
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