Problem 75 - College Algebra with Modeling and Visualization

Question

Solve for the specified variable. $$ \boldsymbol{K}=\frac{1}{2} m \boldsymbol{v}^{2} \text { for } \boldsymbol{v} $$

Step-by-Step Solution

Answer

\( v = \sqrt{\frac{2K}{m}} \)

Step 1: Identify the Known Variables and Rearrange the Equation

The given formula is the kinetic energy formula: \( K = \frac{1}{2}mv^2 \). We need to solve for \( v \). First, start by multiplying both sides of the equation by 2 to get rid of the fraction: \( 2K = mv^2 \).

Step 2: Isolate \( v^2 \)

To isolate \( v^2 \), divide both sides of the equation by \( m \): \( v^2 = \frac{2K}{m} \).

Step 3: Solve for \( v \) by Taking the Square Root

To solve for \( v \), take the square root of both sides of the equation: \( v = \sqrt{\frac{2K}{m}} \). This isolates \( v \) completely.

Key Concepts

Kinematic Equations

In physics, kinematic equations are used to describe the motion of objects. They relate variables of motion such as displacement, velocity, acceleration, and time. Kinetic energy, which we encountered in this exercise, is actually a part of these broader kinematic studies. The kinetic energy equation, \( K = \frac{1}{2} mv^2 \), where \( K \) is kinetic energy, \( m \) is mass, and \( v \) is velocity, specifically describes the energy an object has due to its motion.
Understanding the role of each variable is crucial. The velocity \( v \) is what we're solving for, and recognizing its relationship with kinetic energy helps you connect this equation with others in kinematics.
In practice, kinematic equations require substituting known values and rearranging to find unknowns. The kinetic energy equation is just one example where an understanding of kinematic principles can aid in solving for motion-related variables.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate and solve for a specific variable. It's a fundamental skill in mathematics and science, used extensively in solving physics problems such as those involving kinematic equations.
Let's break down the steps:

• First, you identify known variables in the equation you're given. For us, we have \( K = \frac{1}{2} mv^2 \), where \( K \) and \( m \) are known.
• Next, you need to eliminate any fractions to simplify the equation. Multiply both sides by 2 to get: \( 2K = mv^2 \).
• Finally, rearrange the equation through division by \( m \) to isolate \( v^2 \): \( v^2 = \frac{2K}{m} \).

Using algebraic techniques, you systematically simplify and refactor the equation, preparing it for further steps.

Solving for a Variable

Once the equation is in a format that isolates the squared variable, the next step is solving for the variable itself. Here, we're solving for \( v \), the velocity.
To achieve this, you continue by taking the square root of both sides. This is because you're working with \( v^2 \), and you need to "undo" the squaring. So, \( v = \sqrt{\frac{2K}{m}} \).
This step is crucial in finding the exact value of the variable you're interested in. It requires precision and accuracy.
Checking your work ensures that the algebraic manipulation was correctly done and that you've considered the correct physical principles, consistently applying mathematical operations.