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Chapter 11: Problem 12

Solve the formula for the given variable. $$s=a(x-v t) ; t \quad \text { (Physics) }$$

Short Answer

Expert verified
The solution is \(t = \frac{a \cdot x - s}{a \cdot v}\)

Step by step solution

01

Move terms not involving \(t\) to another side

Start by isolating terms with \(t\) on one side and everything else on the other: \(s=a \cdot x - a \cdot v \cdot t\)
02

Solve for \(t\)

Next, isolate \(t\) by factoring it from the equation and division: \[a \cdot v \cdot t = a \cdot x - s\] By dividing the equation by \(a \cdot v\), we get \[t = \frac{a \cdot x - s}{a \cdot v}\]

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Key Concepts

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

Formula Manipulation
Formula manipulation refers to the process of transforming equations to make them easier to work with or solve. It's like rearranging a puzzle to see the whole picture clearly. When you manipulate a formula, the goal can be to solve for a specific variable or to simplify the equation, making it more understandable.
To effectively manipulate a formula, you can perform several operations on both sides of the equation as long as the equality remains intact. Here are some techniques commonly used:
  • Add or Subtract: You can add or subtract the same term from both sides to get rid of specific terms.
  • Multiply or Divide: This helps to simplify fractions or to isolate variables. Always remember to do it on both sides of the equation.
  • Factor: This refers to the process of taking out common terms or simplifying parts of an equation.
Each of these operations is used flexibly until you reach a form that's more useful for your purpose, such as isolating variables.
Isolating Variables
Isolating variables is a core step in solving equations, especially when interested in a particular parameter out of several. The aim is to get the variable of interest alone on one side of the equation. Let's break it down using a simple method:
  • Identify the Target Variable: Determine which variable you need to isolate.
  • Move Terms: Start rearranging the equation by moving all terms that do not include the variable to the opposite side. Use addition, subtraction, multiplication, or division as necessary.
  • Factor if Needed: If the variable appears in more than one term, try factoring it out to make it easier to isolate.
  • Solve and Simplify: Divide or multiply to finally get the variable alone, simplifying the equation to its cleanest form.
In the context of physics, isolating a variable can help determine a specific physical quantity like time, speed, or distance. By isolating variables, you can better understand how each factor contributes to the outcomes in real-world problems.
Physics Equations
Physics equations describe relationships between different physical quantities. They are foundational tools in understanding the laws of the universe. Here's how to effectively work with them:
  • Understand the Equation: Grasp what each symbol and term represents. Physics formulas often include constants and variables that depict measurable quantities.
  • Apply Correct Units: Ensure that you are using consistent units across the equation to avoid calculation errors.
  • Use Dimensional Analysis: This technique ensures the output of your calculations aligns with expected measurement units.
  • Check Relationships: Understanding how changes in one variable affect others can give you insight into the dynamics of the system being modeled.
Working with physics equations often involves manipulating formulas and isolating variables, as seen in formula manipulation and isolating variables. By mastering these skills, you gain the ability to analyze complex problems, offering a predictive understanding of physical phenomena. Each time you isolate a variable or correctly manipulate a physics formula, you're one step closer to uncovering the mysteries governed by the laws of physics.

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Most popular questions from this chapter

Simplify. $$\frac{x+2}{x y}-\frac{3 x-2}{x^{2} y}$$

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A twin-engine plane flies \(800 \mathrm{mi}\) in the same amount of time it takes a single-engine plane to fly 600 mi. The rate of the twin-engine plane is 50 mph faster than the rate of the single-engine plane. Find the rate of the twin-engine plane.

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