Gravitation

The minimum speed at which an object is thrown to escape the gravitational pull of the moon is known as the escape velocity. The escape velocity is given by the formula: Where:
is the gravitational constant ( ),
is the mass of the moon ( ),
is the radius of the moon ( ).
Substituting the known values: Thus, the minimum speed required is approximately .

The correct option is (E) :

Weight is the force exerted by gravity on an object. The weight of a body is given by the formula: where is the mass of the object and is the acceleration due to gravity. However, when a body is in free fall, it experiences a state of weightlessness. This is because the force of gravity acting on the body and the acceleration due to gravity are balanced, resulting in no normal force acting on the body.
Therefore, the effective weight of the body is zero while in free fall.

The correct option is (E) : zero

The speed of a satellite in a circular orbit is given by the formula: Where: - is the gravitational constant, - is the mass of the planet, - is the radius of the orbit. Let the speed of satellite A be and the speed of satellite B be . For satellite A, the radius of orbit is , and for satellite B, the radius is . From the formula, we know that the speed of satellite A is: Now, using the same formula for satellite B: Using the ratio of the speeds: Therefore:
Thus, the speed of satellite B is .

The correct option is (B) :

Given:

• Escape speed,
• Radius of planet,

Step 1: Relate Escape Speed to Gravity

Escape speed is given by:

where is the gravitational constant and is the planet's mass.

Step 2: Express Surface Gravity

Acceleration due to gravity at the surface ( ) is:

Step 3: Solve for Using Escape Speed

Square the escape speed equation:

Substitute into the gravity formula:

Plug in the given values:

Conclusion:

The acceleration due to gravity at the planet's surface is 5 m/s .

Answer:

The kinetic energy ( ) of a satellite in orbit is given by the formula: where is the mass of the satellite and is the velocity. For a satellite in a circular orbit, the velocity is related to the gravitational force providing the centripetal force: where is the gravitational constant, is the mass of the Earth, and is the radius of the orbit. Solving for : Now, the kinetic energy of a satellite becomes: Let and be the kinetic energies of satellites A and B, respectively. According to the given condition: Substituting the expression for kinetic energy: Simplifying: Thus, the ratio of their masses is:

Step 1: The gravitational force between two masses is given by Newton's Law of Gravitation: where and are the masses and is the separation.
Step 2: The masses of spheres are proportional to their volumes: Step 3: Since they are in contact, the separation is approximately the sum of their radii:
Step 4: Substituting in the gravitational force formula:
Step 5: Simplifying:
Step 6: Since force is inversely proportional to , we get:
Step 7: Therefore, the correct answer is (E). \bigskip