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Chapter 2: Problem 57

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

Short Answer

Expert verified
The solution for \( m \) is \( m = \frac{2K}{v^2} \).

Step by step solution

01

Identify the formula

The formula given is for kinetic energy: \[ K = \frac{1}{2} m v^2 \] Our task is to solve this formula for the variable \( m \).
02

Isolate the term containing m

To isolate \( m \), multiply both sides of the equation by 2 to eliminate the fraction:\[ 2K = m v^2 \]
03

Solve for m

Divide both sides by \( v^2 \) to solve for \( m \):\[ m = \frac{2K}{v^2} \]

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Key Concepts

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

Kinetic Energy Formula
The kinetic energy formula is a key equation in physics that helps us understand how the energy of motion is calculated. Kinetic energy is the energy an object possesses due to its motion and is given by the formula: \[ K = \frac{1}{2} m v^2 \] Here,
  • \( K \) represents the kinetic energy.
  • \( m \) is the mass of the object in kilograms.
  • \( v \) is the velocity of the object in meters per second, squared.
This formula shows that the kinetic energy is directly proportional to both the mass and the square of the velocity of the moving object. Because velocity is squared, even small increases in velocity can lead to significant increases in kinetic energy.
Isolating Variables
Isolating variables is a fundamental algebraic manipulation that involves rearranging an equation so a specific variable stands alone on one side of the equation. This technique is crucial when you're required to solve equations for a particular unknown.
To effectively isolate a variable, you usually follow these steps:
  • Identify the term containing the variable you want to isolate.
  • Use inverse operations to move other terms away from the targeted variable.
  • Simplify the equation by performing any necessary arithmetic operations.
In the given exercise, the goal was to solve the kinetic energy formula for the variable \( m \). By multiplying both sides of the equation by 2, you eliminate the fraction, and the next step is to further simplify by dividing by \( v^2 \). Isolating variables requires careful attention to maintaining the equation's balance through every adjustment.
Solving for a Variable
Solving for a variable involves finding the value or expression of that variable in terms of other known quantities in an equation. It is one step beyond isolating a variable, where you derive an explicit solution for it.
In the given problem, once the term containing \( m \) was isolated: \[ 2K = m v^2 \] The next step involved solving for \( m \) by dividing both sides of the equation by \( v^2 \). This resulted in: \[ m = \frac{2K}{v^2} \]
  • This manipulation provides the mass \( m \) as the subject of the formula in terms of kinetic energy \( K \) and velocity \( v \).
  • It's a valuable skill in mathematics and physics as it allows you to express unknowns in a way that can be calculated or further analyzed.
Solving for a variable equips you with the ability to transform formulas and apply them to varied contexts, helping you find solutions in a wide range of problems.

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