The kinetic energy, K, of an object is given by the formula K = 1/2mv^2, where m is the mass...

1. TRANSLATE the problem information

• Given information:
  • Kinetic energy formula: \(\mathrm{K = \frac{1}{2}mv^2}\)
  • Satellite S has \(\mathrm{K = 36,000}\) joules
  • New satellite has half the mass: \(\mathrm{\frac{m}{2}}\)
  • New satellite has three times the speed: \(\mathrm{3v}\)

2. INFER the most efficient approach

• Rather than trying to find specific values of m and v, we can express the new kinetic energy in terms of the original kinetic energy
• This lets us work directly with the given value of 36,000 joules

3. Set up the original satellite equation

Let satellite S have mass m and speed v:

\(\mathrm{K_{original} = \frac{1}{2}mv^2 = 36,000}\) joules

4. TRANSLATE and set up the new satellite equation

\(\mathrm{K_{new} = \frac{1}{2} \times \frac{m}{2} \times (3v)^2}\)

5. SIMPLIFY the new equation step by step

• First, handle the exponent: \(\mathrm{(3v)^2 = 9v^2}\)
• \(\mathrm{K_{new} = \frac{1}{2} \times \frac{m}{2} \times 9v^2}\)
• \(\mathrm{K_{new} = \frac{1}{2} \times \frac{1}{2} \times 9 \times mv^2}\)
• \(\mathrm{K_{new} = \frac{9}{4}mv^2}\)

6. INFER the relationship to original kinetic energy

Since \(\mathrm{K_{original} = \frac{1}{2}mv^2 = 36,000}\), we can write:

\(\mathrm{K_{new} = \frac{9}{4}mv^2}\)

\(\mathrm{K_{new} = \frac{9}{2} \times [\frac{1}{2}mv^2]}\)

\(\mathrm{K_{new} = \frac{9}{2} \times 36,000}\)

7. Calculate the final answer

\(\mathrm{K_{new} = 4.5 \times 36,000 = 162,000}\) joules

Answer: C. 162,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly handle the speed modification, treating "three times the speed" as just \(\mathrm{3v}\) in the final calculation instead of recognizing it needs to be squared as \(\mathrm{(3v)^2 = 9v^2}\).

This leads them to calculate:

\(\mathrm{K_{new} = \frac{1}{2} \times \frac{m}{2} \times 3v}\)

\(\mathrm{K_{new} = \frac{3}{4}mv^2}\)

\(\mathrm{K_{new} = \frac{3}{2} \times 36,000 = 54,000}\)

This may lead them to select Choice A (54,000).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{K_{new} = \frac{9}{4}mv^2}\) but make arithmetic errors when converting this to terms of the original kinetic energy, such as incorrectly calculating \(\mathrm{\frac{9}{4} \div \frac{1}{2}}\) or mishandling the final multiplication.

This leads to confusion and potentially selecting Choice B (108,000) or Choice D (324,000) depending on their specific calculation error.

The Bottom Line:

This problem tests whether students can systematically apply the kinetic energy formula with modified parameters while carefully handling both the translation of verbal descriptions into mathematical expressions and the algebraic manipulation of those expressions.