The formula E = 1/2mv^2 gives the kinetic energy E of an object with mass m moving at speed v....

1. INFER the solution strategy

• Goal: Isolate the variable v on one side of the equation
• Strategy: Undo operations affecting v in reverse order
• Current situation: v is squared, multiplied by m, multiplied by ½

2. SIMPLIFY by eliminating the fraction first

• Start with: \(\mathrm{E = \frac{1}{2}mv^2}\)
• Multiply both sides by 2: \(\mathrm{2E = mv^2}\)
• This removes the fraction and makes subsequent steps cleaner

3. SIMPLIFY by isolating v²

• From: \(\mathrm{2E = mv^2}\)
• Divide both sides by m: \(\mathrm{\frac{2E}{m} = v^2}\)
• Now v² is isolated on the right side

4. SIMPLIFY by taking the square root

• From: \(\mathrm{\frac{2E}{m} = v^2}\)
• Take square root of both sides: \(\mathrm{v = \sqrt{\frac{2E}{m}}}\)
• This gives us v isolated completely

Answer: A. v = \(\mathrm{\sqrt{\frac{2E}{m}}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly multiply by 2 but then make an error in the fraction manipulation, thinking they should divide E by 2m instead of multiplying E by 2 first.

Their work might look like:

\(\mathrm{E = \frac{1}{2}mv^2}\)

\(\mathrm{v^2 = \frac{E}{\frac{1}{2}m}}\)

\(\mathrm{v^2 = \frac{E}{2m}}\)

\(\mathrm{v = \sqrt{\frac{E}{2m}}}\)

This may lead them to select Choice B (\(\mathrm{\sqrt{\frac{E}{2m}}}\))

Second Most Common Error:

Poor INFER reasoning: Students correctly eliminate the fraction and isolate v², but forget that they need to take the square root to solve for v (not v²).

Their work stops at: \(\mathrm{\frac{2E}{m} = v^2}\), and they think this means \(\mathrm{v = \frac{2E}{m}}\)

This may lead them to select Choice C (\(\mathrm{\frac{2E}{m}}\))

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires carefully undoing each operation that affects the target variable v, in the proper reverse order, while maintaining algebraic accuracy throughout.