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Chapter 1: Problem 78

Solve for the indicated variable. $$ t=\sqrt{\frac{2 s}{g}} \text { for } s $$

Short Answer

Expert verified
s = \frac{t^2 g}{2}

Step by step solution

01

Isolate the square root

First, isolate the square root expression on one side of the equation. The given equation is: \[ t = \sqrt{\frac{2 s}{g}} \]This step is complete because the square root is already isolated.
02

Remove the square root

Square both sides of the equation to remove the square root. \[ t^2 = \left( \sqrt{\frac{2s}{g}} \right)^2 \]Simplify to get: \[ t^2 = \frac{2s}{g} \]
03

Solve for the variable

Multiply both sides of the equation by \( g \) to isolate \( s \). \[ t^2 \times g = \frac{2s}{g} \times g \]This simplifies to: \[ t^2 g = 2s \]Divide both sides by 2 to solve for \( s \). \[ s = \frac{t^2 g}{2} \]

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Key Concepts

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

algebraic manipulation
Algebraic manipulation refers to the process of rearranging and simplifying equations or expressions. This is a fundamental skill in solving equations, and it consists of several basic operations:
  • Adding or subtracting the same quantity from both sides of an equation
  • Multiplying or dividing both sides of an equation by the same non-zero quantity
  • Using the distributive, associative, and commutative properties to simplify expressions
Let鈥檚 understand this with an example from our problem: We started with the equation where \(t\) represents time, \(s\) represents the displacement, and \(g\) represents acceleration due to gravity: \[t = \sqrt{\frac{2 s}{g}}\r\] In the second step, we squared both sides to remove the square root, leading to: \[ t^2 = \frac{2s}{g} \r\] Performing these operations correctly is key to solving equations.
isolating variables
Isolating variables means rearranging an equation so that the variable you want to solve for is by itself on one side of the equation. This is crucial in deriving the required result from a formula or expression.
To isolate the variable \(s\) in the given exercise, we performed the following steps:
1. We squared both sides of the original equation to remove the square root:
\[ t^2 = \frac{2s}{g} \r\]
2. Then, we multiplied both sides by \(g\) to eliminate the fraction: \[ t^2 \times g = 2s \r\]
3. Finally, we divided both sides by 2 to solve for \(s\). \[ s = \frac{t^2 g}{2} \r\]
Each operation was aimed towards getting \(s\) alone on one side of the equation.
quadratic equations
Quadratic equations are polynomial equations of degree 2. They have the general form: \[ ax^2 + bx + c = 0 \r\] Where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
Although the original equation in our exercise was not a quadratic equation, the algebraic manipulation process led to a form similar to quadratic expressions after squaring both sides. The final form, \[ s = \frac{t^2 g}{2} \r\] Simplifies solving for \(s\), indicating practical use of quadratic properties in more complex derivations. Recognizing when to apply rules from quadratic equations is a key skill whenever you're dealing with squared terms.

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