- Multiply both sides by 2 to get \(2E = mv^2\).
- Divide by \(m\) to obtain \(v^2 = \frac{2E}{m}\).
- Take the square root of both sides, resulting in \(v = \pm \sqrt{\frac{2E}{m}}\).
- Final answer: \(\boxed{v = \pm \sqrt{\frac{2E}{m}}}\).
Explanation
- Understanding the Formula and Objectives
The problem asks us to solve the physical formula for kinetic energy, \(E = \frac{1}{2} m v^2\), for the velocity \(v\). Here, \(E\) represents the energy, \(m\) denotes the mass, and \(v\) is the velocity. To isolate \(v\), we must perform a series of algebraic operations to shift all other variables to the other side of the equation.
- Algebraic Manipulation to Isolate Velocity
Starting with the given formula:
\[E = \frac{1}{2} m v^2\]
Our first goal is to eliminate the fraction. We multiply both sides of the equation by 2:
\[2E = m v^2\]
Next, to isolate \(v^2\), we divide both sides by \(m\) (assuming \(m
eq 0\)):
\[v^2 = \frac{2E}{m}\]
Finally, we solve for \(v\) by taking the square root of both sides. Since \(v\) could be either positive or negative, we include both signs:
\[v = \pm \sqrt{\frac{2E}{m}}\]
- Conclusion and Final Formula
Comparing our derived formula \(v = \pm \sqrt{\frac{2E}{m}}\) with the provided options:
- \(v = \pm \sqrt{\frac{m}{2E}}\)
- \(v = 2E - m\)
- \(v = \pm \sqrt{2E - m}\)
- \(v = \pm \sqrt{\frac{2E}{m}}\)
The correct formula is \(v = \pm \sqrt{\frac{2E}{m}}\).
Examples
In physics, knowing the velocity of an object from its energy and mass is essential. For example, if a car of mass \(m\) has a certain kinetic energy \(E\), you can use the formula \(v = \pm \sqrt{\frac{2E}{m}}\) to determine how fast it was moving. This conversion is a fundamental skill in mechanics for solving problems related to motion and energy conservation.