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The problem asks for the weight of a person on a new planet based on that person's weight on Earth.

### Step-by-step explanation:

1. **Weight on a Planet:**
   Weight \( W \) is determined by the gravitational force, which depends on both the mass of the planet and the distance from its center (radius). Mathematically, weight \( W \) on a planet's surface is given by the formula:

   \[
   W = \frac{G M_p m}{R_p^2}
   \]
   where:
   - \( G \) is the gravitational constant,
   - \( M_p \) is the mass of the planet,
   - \( m \) is the mass of the person,
   - \( R_p \) is the radius of the planet.

2. **Comparison with Earth's Weight:**
   For the Earth, weight \( W_{\text{Earth}} \) is:

   \[
   W_{\text{Earth}} = \frac{G M_{\text{Earth}} m}{R_{\text{Earth}}^2}
   \]
   
   Now, let's define the new planet's mass and radius:
   - The mass of the new planet is **half** that of Earth's mass: \( M_p = \frac{M_{\text{Earth}}}{2} \).
   - The radius of the new planet is **twice** Earth's radius: \( R_p = 2 R_{\text{Earth}} \).

3. **Weight on the New Planet:**
   Substituting the values of \( M_p \) and \( R_p \) into the weight equation for the new planet:

   \[
   W_{\text{new}} = \frac{G \cdot \frac{M_{\text{Earth}}}{2} \cdot m}{(2 R_{\text{Earth}})^2}
   \]

   Simplifying:

   \[
   W_{\text{new}} = \frac{G M_{\text{Earth}} m}{2 \cdot 4 R_{\text{Earth}}^2} = \frac{1}{8} \cdot \frac{G M_{\text{Earth}} m}{R_{\text{Earth}}^2}
   \]

   Since \( \frac{G M_{\text{Earth}} m}{R_{\text{Earth}}^2} = W_{\text{Earth}} \), we get:

   \[
   W_{\text{new}} = \frac{W_{\text{Earth}}}{8}
   \]

   Therefore, the weight of the person on the new planet is **\( \frac{w}{8} \)**, which corresponds to option **(a)**.

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### Related Questions:
1. How is the gravitational force calculated between two objects?
2. How does the mass of a planet affect a person's weight?
3. What would happen to the weight if the radius was three times Earth's radius instead of two?
4. How does weight vary with the square of the radius?
5. What is the relationship between mass and gravity on different celestial bodies?

### Tip:
Always remember that gravitational force depends on both mass and the inverse square of the distance (radius), so even small changes in radius can greatly affect weight.

A person who has weight w on Earth's surface travels to a remote planet. That new planet has a mass that is half of the Earth's mass and a radius that is twice Earth's radius. What is the weight of the person close to the new planet's surface?

Discover how to calculate the weight of a person on a new planet with half the mass of Earth and twice its radius. This problem involves applying Newton's Law of Universal Gravitation and the inverse square law. Learn the key concepts and formulas needed to solve for weight in a gravitational field. Suitable for high school physics students.

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