The force of attraction of the Earth to the Sun is 22.5 times greater than the force of attraction of Mars to the Sun. How many times is the distance between Mars and the Sun greater than the distance between the Earth and the Sun if the mass of the Earth is 10 times the mass of Mars?
Forces Fz and Fm, with which the Earth and Mars are attracted to the Sun:
Fz = (G * Mz * Ms) / (Rz) ² (1);
Fm = (G * Mm * Ms / (Rm) ² (2).
Designations in formulas:
G – gravitational constant,
Mz is the mass of the Earth,
Mm is the mass of Mars,
MS is the mass of the Sun,
Rz is the distance between the Earth and the Sun,
Rm is the distance between Mars and the Sun.
Let’s rewrite (1), taking into account the relation Mz = 10 * Mm.
Fz = (G * 10 * Mm * Ms) / (Rz) ² (3);
By condition Fz = 22.5 * Fm. We substitute instead of Fz and Fm expressions from (3) and (2):
(G * 10 * Mm * Ms) / (Rz) ² = 22.5 * (G * Mm * Ms / (Rm) ².
Reducing the factors:
10 / (Rz) ² = 22.5 / (Rm) ²;
10 * (Rm) ² = 22.5 * (Rз) ²;
Rm = root (22.5 / 10) * Rz;
Rm = 1.5 * Rz.
Answer: 1.5 times.
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