Problem 16 Solve each formula for the quant... [FREE SOLUTION]

Chapter 2: Problem 16

Solve each formula for the quantity given. $$ E_{k}=\frac{1}{2} m v^{2} \text { for } v^{2} $$

Short Answer

Expert verified

$ v^2 = \frac{2E_k}{m} $

Step by step solution

Identify the given equation

The given formula is the expression for kinetic energy: $ E_k = \frac{1}{2} mv^2 $. We are asked to solve for $ v^2 $.

Isolate $ v^2 $

To isolate $ v^2 $, we need to remove any coefficients or factors applied to it. The formula is $ E_k = \frac{1}{2} mv^2 $. Begin by multiplying both sides by 2 to get rid of the fraction: $ 2E_k = mv^2 $.

Solve for $ v^2 $

Now, divide both sides by $ m $ to solve for $ v^2 $: $ \frac{2E_k}{m} = v^2 $. This isolates $ v^2 $ on one side of the equation.

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Key Concepts

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

Physics Formula Manipulation

Physics often requires manipulating formulas to solve for a specific variable. This skill enables the modification of equations to better suit the problem at hand. Let's consider the provided kinetic energy equation.

The kinetic energy ($E_k$) equation is given by:\[E_k = \frac{1}{2} mv^2\], where$m$ is mass and$v$ is velocity.
When asked to solve for$v^2$, start by determining which operations need to be reversed.

Manipulating the formula involves strategically using arithmetic operations to rearrange it, while maintaining balance. It鈥檚 like solving for an unknown variable using basic algebra. The goal is to isolate the desired quantity (here,$v^2$) on one side of the equation. In this case, we need to eliminate fractions and coefficients by performing the inverse operation.

Kinetic Energy Equation

The kinetic energy equation provides the framework for understanding how an object's motion relates to its mass and velocity. It is expressed as:\[E_k = \frac{1}{2} mv^2\]. This tells us:

Kinetic energy ($E_k$) is the energy possessed by an object due to its motion.
The factor$\frac{1}{2}$ arises from the integration of the potential energy in classical mechanics.
Mass ($m$) represents how much matter is in the object.
Velocity ($v$) is the speed of the object in a given direction.

Understanding this equation is crucial because it can be rearranged to find other variables, such as mass or velocity squared, depending on what information is available. This manipulation highlights the versatility of the formula in practical physics scenarios.

Solving for Velocity Squared

Solving for velocity squared ($v^2$) means isolating this particular term from the kinetic energy equation. Here鈥檚 how:

Start with the initial kinetic energy equation:\[E_k = \frac{1}{2} mv^2\].
To eliminate the fraction, multiply both sides by 2:\[2E_k = mv^2\].
Finally, to isolate$v^2$, divide both sides by the mass$m$:\[v^2 = \frac{2E_k}{m}\].

This manipulation allows us to directly calculate velocity squared if kinetic energy and mass are known. It's a clear example of using arithmetic operations to rearrange and simplify equations, making it easier to find unknowns in physics problems.

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91影视

Solve each formula for the quantity given. $$ E_{k}=\frac{1}{2} m v^{2} \text { for } v^{2} $$

Short Answer

Step by step solution

Identify the given equation

Isolate \( v^2 \)

Solve for \( v^2 \)

Key Concepts

Physics Formula Manipulation

Kinetic Energy Equation

Solving for Velocity Squared

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