When a spacecraft is launched from the earth toward the sun,at what distance from the earth will the gravitational forces due to the sun and the earth cancel? Earth's mass is 5.97 × 10²⁴ kg,the sun's mass is 1.99 × 10³⁰ kg,and the Earth-sun distance is 1.5 × 10¹¹ m. A)1.3 × 10⁸ m B)2.6 × 10⁸ m C)1.3 × 10¹⁰ m D)2.6 × 10¹⁰ m

Question

Quizplus · Accepted Answer

The gravitational force exerted on the spacecraft by the sun and the earth are given by

Fsun = Gmsunm_spacecraft/r^2          (1)

Fearth = Gmearthm_spacecraft/(R-r)^2      (2)

where G is the universal gravitational constant and r is the distance from the center of the earth to the spacecraft (the distance from the center of the sun to the spacecraft is R = 1.5 × 10^11 m).

At the point where the two forces cancel out, Fsun = Fearth.

Therefore, we have

Gmsunm_spacecraft/r^2 = Gmearthm_spacecraft/(R-r)^2

Simplifying this equation yields r = (mearth/mtotal)R = (5.97×10^24 kg)/(5.97×10^24 + 1.99×10^30 kg) × 1.5×10^11 m = 2.6 × 10^18 m.

Therefore, the best answer is (B) 2.6 × 10^18 m.

When a Spacecraft Is Launched from the Earth Toward the Sun,at