Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(s=\frac{1}{2} g t^{2}+v_{0} f\) for \(v_{0}\)

Step-by-Step Solution

Answer

The solution for \( v_0 \) is \( v_0 = \frac{s - \frac{1}{2}gt^2}{f} \).

Step 1: Understand the Formula

The formula given is \( s = \frac{1}{2}gt^2 + v_0f \). It involves the variables \( s \), \( g \), \( t \), \( v_0 \), and \( f \). We want to isolate and solve for \( v_0 \).

Step 2: Identify the Terms

Notice that the formula consists of two main terms on the right-hand side: \( \frac{1}{2}gt^2 \) and \( v_0f \). To solve for \( v_0 \), we need to isolate this term.

Step 3: Rearrange the Equation

Subtract \( \frac{1}{2}gt^2 \) from both sides of the equation to isolate the term involving \( v_0 \). This gives us \( s - \frac{1}{2}gt^2 = v_0f \).

Step 4: Solve for \( v_0 \)

Divide both sides by \( f \) to solve for \( v_0 \). So, \( v_0 = \frac{s - \frac{1}{2}gt^2}{f} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental technique used to solve equations and involves the rearrangement of terms in an equation to make it easier to work with. It is the process of applying mathematical operations, such as addition and subtraction, and applying properties like the distributive property to transform an equation into a desired form.
To manipulate algebraic equations, you often aim to simplify complex expressions into more manageable components. This involves:

• Combining like terms to condense expressions.
• Moving terms across the equal sign using opposite operations (e.g., adding if a term is subtracted).
• Applying multiplication or division to both sides to balance the equation.

A good tip is to perform the same operation on both sides of the equation to maintain equality. This balance is crucial in ensuring that the equation remains valid after manipulation. Practicing these techniques can significantly improve problem-solving abilities in mathematics.

Formula Rearrangement

Formula rearrangement is a strategic method used in mathematics to solve for one variable within a formula, given the values of other variables. This process requires a deep understanding of mathematical operations and properties to correctly rearrange the given formula without altering its meaning.
The key to successful formula rearrangement is carefully identifying the target variable and systematically moving other terms away from it. Here’s how you can effectively rearrange a formula:

• Identify what you are solving for: Focus on the variable that needs isolation.
• Use inverse operations: If terms are added, subtract them from both sides to remove them. Similarly, use division if a term multiplies the variable.
• Maintain balance: Ensure that each step keeps the equation balanced by performing equal operations on both sides.

In solving the original exercise, the goal is to isolate \(v_0\). This involves carefully moving other terms by subtraction and division, showcasing a clear example of formula rearrangement.

Variable Isolation

Variable isolation is a crucial technique in solving equations that involves "unwrapping" the variable of interest so that it stands alone on one side of the equation. This specific process is essential when you're required to solve for a single variable in a complex equation.
Steps in variable isolation include:

• Identify the variable to be isolated: Clearly determine which variable is the subject of the equation.
• Eliminate coefficients: If the variable is multiplied by a coefficient, use division to eliminate this coefficient.
• Shift terms: Use mathematical operations to shift other terms across the equal sign.

In the case of our exercise, the variable \(v_0\) is isolated by eliminating all other terms adjacent to it on the equation, using subtraction and division. This process makes \(v_0\) the subject of the formula, ready to be calculated once the values of \(s\), \(g\), \(t\), and \(f\) are known. Mastery of variable isolation equips you with the tools to solve a wide array of mathematical problems.