Problem 88 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Kinetic Energy Solve for \(m\) in \(E=\frac{1}{2} m v^{2}\).

Step-by-Step Solution

Answer

The formula for \(m\) in terms of \(E\) and \(v\) is \(m = \frac{2E}{v^{2}}\).

Step 1: Analyze the formula

The formula given is \(E=\frac{1}{2} m v^{2}\). In this formula, \(E\) is the kinetic energy, \(m\) is the mass and \(v\) is the velocity.

Step 2: Rearrange the formula

We want to isolate \(m\), so we have to eliminate \(\frac{1}{2}\) and \(v^{2}\) from the right side of the equation. We can do this by multiplying both sides of the equation by \(\frac{2}{v^{2}}\). This gives us \(\frac{2E}{v^{2}} = m\).

Step 3: Simplify the formula

We now have the formula \(\frac{2E}{v^{2}} = m\), where \(m\) is isolated as required. This formula can be used to calculate the mass of the object if you know the kinetic energy and velocity.

Key Concepts

Kinetic Energy Equation

Kinetic energy is the energy that an object possesses due to its motion. It's an essential concept in physics that plays a vital role in both theoretical and practical applications. The standard equation for kinetic energy (\(E\text{K}\)) is expressed as \(E=\frac{1}{2} m v^{2}\), where \(E\) represents the kinetic energy, \(m\) is the mass of the object, and \(v\) is its velocity or speed.

This equation tells us that the kinetic energy of an object is directly proportional to the mass and the square of its velocity. This means that even a small increase in the velocity of the object will have a significant impact on its kinetic energy because of the squaring of \(v\). In practical terms, this is why the impact of a fast-moving vehicle is much more severe than one that is moving slowly, even if their masses are the same.

Rearranging Algebraic Formulas

Mastering the skill of rearranging algebraic formulas is crucial in solving for variables in physics as well as in other scientific domains. It involves the manipulation of the equation to solve for a particular variable. This process generally includes operations such as addition, subtraction, multiplication, division, and 'taking roots', which are applied to both sides of the equation to maintain equilibrium.

For instance, in the kinetic energy equation, if we wish to solve for mass (\(m\)), we need to rearrange the formula in a way that \(m\) is isolated on one side. The art of rearranging becomes even more critical when dealing with more complex equations involving multiple variables and operations.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable one wants to solve for is on one side of the equation and everything else is on the other side. In other words, you're 'solving for' that particular variable. In the context of the kinetic energy equation, solving for the mass \(m\) meant we needed to isolate \(m\) from other variables and constants.

To do this, we divided both sides by \(v^{2}\) and then multiplied by 2, effectively performing the opposite operations to eliminate the fraction and the square on the right-hand side. As we went through the steps methodically, we successfully isolated the variable, which now reads \(m=\frac{2E}{v^{2}}\). This simplified form makes it straightforward to calculate the mass when the kinetic energy and velocity are known.