## Q. 7.10

CERES

GOAL Relate Newton’s universal law of gravity to $m g$ and show how $g$ changes with position.

PROBLEM An astronaut standing on the surface of Ceres, the largest asteroid, drops a rock from a height of $10.0 \mathrm{~m}$. It takes $8.06 \mathrm{~s}$ to hit the ground. (a) Calculate the acceleration of gravity on Ceres. (b) Find the mass of Ceres, given that the radius of Ceres is $R_C=5.10 \times 10^2 \mathrm{~km}$. (c) Calculate the gravitational acceleration $50.0 \mathrm{~km}$ from the surface of Ceres.

STRATEGY Part (a) is a review of one-dimensional kinematics. In part (b) the weight of an object, $w=m g$, is the same as the magnitude of the force given by the universal law of gravity. Solve for the unknown mass of Ceres, after which the answer for (c) can be found by substitution into the universal law of gravity, Equation 7.21.

$F = G \frac{m_1 m_2}{r^2}$      [7.21]

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