Question

A spacecraft goes from the earth directly to the sun. How far from the centre of the earth the gravitational forces exerted on it by the earth and by the sun would be of equal magnitude? The distance between the earth and the sun is 1.49 × 10⁸ km. The masses of the sun and the earth are 2 × 10³⁰ kg and 6 × 10²⁴ kg respectively.

Answer 1

Answer:

$2.62 \times {10}^{5} km$

Explanation:

Refer to the material.

Answer 2

Answer:

$2.57\times 10^5\ km$

Explanation:

Let's the mass of the spacecraft is m and at a distance x the gravitational force exerted by moon and sun is equal.

Given,

Distance between earth and the sun,d = 1.49 × 10⁸ km

$\textrm{Mass of the sun},M_1=\ 2\times 10^{30}\ kg$

$\textrm{Mass of the earth},\ M_2=\ 6\times 10^{24}\ kg$

gravitational force on the spacecraft can be given by,

$F\ =\ \dfrac{GMm}{R^2}$

So, according to question

$\dfrac{G.M_2.m}{x^2}\ =\ \dfrac{GM_1.m}{(d-x)^2}$

$=>\ \dfrac{M_2}{x^2}\ =\ \dfrac{M_1}{(d-x)^2}$

$=>\ \dfrac{6\times 10^{24}}{x^2}\ =\ \dfrac{2\times 10^{30}}{(1.49\times 10^8\ -\ x)^2}$

$=>\ \dfrac{1.49\times 10^8\ -\ x}{x}\ =\ \dfrac{10^3}{\sqrt{3}}$

$=>\ \dfrac{1.49\times 10^8}{x}-1\ =\ \dfrac{10^3}{\sqrt{3}}$

$=>\ x\ =\ 2.57\times 10^5\ km$

Hence, the spacecraft $2.57\times 10^5\ km$ from the earth's center the gravitational force on the spacecraft due to the sun and the earth will be equal.