An astronaut drops a marble on the surface of the airless Planet Z-49 and observes that it takes 1.02 s for the marble to fall 2.00 m starting from rest. She also knows that the radius of Z-49 is 3.39 × 10⁶ m. From this information, what will she determine for the mass of Z-49? (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 3.30 × $10^{23}$ kg. B) 6.62 × $10^{23}$ kg. C) 4.62 × $10^{23}$ kg. D) 8.09 × $10^{23}$ kg. E) 9.95 × $10^{23}$ kg.

Question

Quizplus · Accepted Answer

Using the formula for the time taken for an object to fall a certain distance from rest, $t=\sqrt{\dfrac{2h}{g}}$, we can solve for the acceleration due to gravity on Planet Z-49:
$$g=\dfrac{2h}{t^2}=\dfrac{2(2.00	ext{ m})}{(1.02	ext{ s})^2}=3.87654... 	ext{ m/s}^2$$
Then, using the formula for the acceleration due to gravity at the surface of a planet, $g=\dfrac{GM}{r^2}$, we can solve for the mass $M$ of Planet Z-49:
$$M=\dfrac{gr^2}{G}=\dfrac{(3.87654...	ext{ m/s}^2)(3.39	imes 10^{16}	ext{ m})^2}{6.67	imes 10^{-11}	ext{ N }\cdot	ext{ m}^2/	ext{kg}^2}=6.62	imes 10^{23}	ext{ kg}$$
Therefore, the answer is B.

An Astronaut Drops a Marble on the Surface of the Airless