An object's kinetic energy, in joules, is equal to the product of one-half the object's mass, in kilograms, and the...

1. TRANSLATE the problem information

• Given information:
  • Kinetic energy = 18 joules
  • Mass = 4 kilograms
  • Need to find: speed in meters per second

• The problem tells us that kinetic energy equals the product of one-half the object's mass and the square of the object's speed

2. TRANSLATE the relationship into formula

• From the word description: \(\mathrm{KE = \frac{1}{2}mv^2}\)
  • KE = kinetic energy
  • m = mass
  • v = speed

3. SIMPLIFY by substituting known values

• Substitute KE = 18 and m = 4:

\(\mathrm{18 = \frac{1}{2}(4)v^2}\)

• This becomes: \(\mathrm{18 = 2v^2}\)

4. SIMPLIFY to solve for v²

• Divide both sides by 2: \(\mathrm{9 = v^2}\)

5. SIMPLIFY to find v

• Take the square root: \(\mathrm{v = ±\sqrt{9} = ±3}\)

6. APPLY CONSTRAINTS to select final answer

• Since speed cannot be negative in real-world context: \(\mathrm{v = 3\text{ m/s}}\)

Answer: A. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "the product of one-half the object's mass and the square of the object's speed" and write incorrect formulas like \(\mathrm{KE = \frac{1}{2}m + v^2}\) or \(\mathrm{KE = \frac{1}{2}(mv)^2}\).

Without the correct formula, they cannot set up the equation properly and end up guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{18 = 2v^2}\) but make algebraic errors. They might forget to divide by 2, leading to \(\mathrm{v^2 = 18}\), so \(\mathrm{v = \sqrt{18} \approx 4.24}\), causing confusion since this isn't an answer choice. Or they might solve correctly to get \(\mathrm{v^2 = 9}\) but forget to take the square root, selecting Choice C (9) thinking this is the final answer.

The Bottom Line:

This problem tests whether students can convert a physics relationship described in words into a mathematical formula and then solve systematically. The key challenge is maintaining precision through each algebraic step while recognizing physical constraints on the final answer.