Let's analyze the given problem step-by-step.
1. Given Information:
- Planets A and B have the same mass.
- Planet A is half the size of planet B.
2. Key Concepts:
- Weight on a planet depends on the gravitational force exerted by the planet.
- Gravitational force [tex]\((F)\)[/tex] is given by Newton's law of universal gravitation:
[tex]\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/tex]
where [tex]\(G\)[/tex] is the gravitational constant, [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are the masses of the two objects (e.g., a person and the planet), and [tex]\(r\)[/tex] is the distance between the centers of the two masses (the radius of the planet if we consider the person on the surface).
- If two planets have the same mass but different sizes, the gravitational force (and thereby the weight experienced) will depend on the radius of the planets.
3. Analysis of Each Statement:
- Statement A: "You would weigh the same on both planets because your mass would be the same on both."
- This is incorrect. Your weight depends on the gravitational force, not just your mass. The force also involves the radius of the planet.
- Statement B: "You would weigh less on planet A because the distance between you and the planet's center of gravity would be smaller."
- This is incorrect. A smaller distance (smaller radius) would actually result in a stronger gravitational force and hence a greater weight.
- Statement C: "You would weigh the same on both planets because the masses of the planets are the same."
- This is incorrect. Even though the masses of the planets are the same, the gravitational force also depends on the radius of the planet, which is different.
- Statement D: "You would weigh more on planet A because the distance between you and the planet's center of gravity would be smaller."
- This is correct. Since planet A is half the size of planet B, the radius (r) is smaller for planet A. A smaller radius increases the gravitational force due to the inverse-square law [tex]\(\left( \frac{1}{r^2} \right)\)[/tex]. Thus, you would weigh more on planet A.
4. Conclusion:
- The correct statement is D: "You would weigh more on planet A because the distance between you and the planet's center of gravity would be smaller."
Therefore, the correct answer is D.
