Question

19 & 20

019 (part 1 of 2) 10.0 points On the way to the moon, the Apollo astro- nauts reach a point where the Moon's gravi- tational pull is stronger than that of Earth's. Find the distance of this point from the center of the Earth. The masses of the Earth and the Moon are 5.98 x 1024 kg and 7.36x 1022 kg, respectively, and the distance from the Earth to the Moon is 3.84 x 108 m Answer im units of m. 020 (part 2 of 2) 10.0 points What would the acceleration of the astro- naut be due to the Earth's gravity at this point if the moon was not there? The value of the universal gravitational constant is 6.672 x 10 11 N m2/kg2 Answer in units of m/s

Answer 1

Given

M_e = 5.98 x 10²⁴ kg

R = 3.84 x 10⁸m

M_m = 7.36 x 10²² kg

Assume if r_e is the distance from this point to the center of the earth and r_m is the distance from this point to the center of the moon

G*m*M_e/(r_e²) = G*m*M_m/(r_m²)……eq1

and

we know

r_e + r_m = R……………eq2

solving for r_e = distance from center on the earth

G*m*M_e/(r_e²) = G*m*M_m/(r_m²)

M_e/(r_e²) = M_m/(r_m²)

r_m / r_e = $\sqrt$(M_m/M_e)

r_m / r_e = $\sqrt$(7.36 x 10²²/5.98 x 10²⁴)

r_m / r_e = 0.1109400392

replacing value for eq 2

r_e = R/(1+0.1109400392)

r_e = (3 x 10⁸)/(1+0.1109400392)

r_e = 2.7 x 10⁸ m

(20)

g' = G*M_e/r_e²

g' = (6.672x10^-11*5.98x10²⁴)/(2.7 x 10⁸)²

g' = 0.0054713 m/s²

g' = 5.47 x 10^-3 m/s²