Question

A rocket is fired from the Earth towards the Sun. At what distance from the Earth's centre, the gravitational force on the rocket is zero? Mass of the Sun = \( 2 \times 10^{30} \, \text{kg} \) and mass of the Earth = \( 6 \times 10^{24} \, \text{kg} \).

• A. \( 2.6 \times 10^8 \, \text{m} \)
• B. \( 3.2 \times 10^8 \, \text{m} \)
• C. \( 3.9 \times 10^9 \, \text{m} \)
• D. \( 2.3 \times 10^9 \, \text{m} \)

Hint:

To find the point where the gravitational forces from two bodies cancel each other out, use the ratio of their masses and distances.

Answer 1

The Correct Option is C

The gravitational force from the Earth and the Sun on the rocket will cancel out at a point where the net gravitational force is zero.
The formula for the gravitational force is: \[ F = \frac{G M_1 M_2}{r^2} \] where \( G \) is the gravitational constant, \( M_1 \) and \( M_2 \) are the masses, and \( r \) is the distance between the objects. For zero gravitational force, the force due to Earth’s gravity and the force due to Sun’s gravity must be equal: \[ \frac{G M_{\text{Earth}} m}{r^2} = \frac{G M_{\text{Sun}} m}{(R - r)^2} \] Simplifying the equation: \[ \frac{M_{\text{Earth}}}{r^2} = \frac{M_{\text{Sun}}}{(R - r)^2} \] where \( R \) is the distance between the Earth and the Sun.
Solving for \( r \), we find: \[ r = 3.9 \times 10^9 \, \text{m} \]
Thus, the correct answer is (c).