You drop an egg from rest with no air resistance.

As it falls,

a. only its momentum is conserved.
b. only its kinetic energy is conserved.
c. both its momentum and its mechanical energy are conserved.
d. its mechanical energy is conserved, but its momentum is not conserved

There is a force on it in the direction of motion. Therefore its kinetic energy increases. However its total mechanical energy including the gravitational potential energy is conserved.

Meanwhile net force causes a rate of change of momentum, called acceleration when the mass is constant. Therefore the momentum changes.

When an egg is dropped from rest with no air resistance, the only force acting on it is gravity. As it falls, the gravitational force causes it to accelerate and gain velocity. In this scenario, both its momentum and its mechanical energy are conserved.

Momentum is defined as the product of an object's mass and its velocity. In this case, as the egg falls, its mass remains constant and its velocity increases. Therefore, its momentum is conserved.

Mechanical energy is the sum of an object's potential energy and kinetic energy. As the egg falls, its potential energy decreases while its kinetic energy increases. However, the total mechanical energy (potential energy + kinetic energy) remains constant, as no other external forces are acting on the egg. Thus, its mechanical energy is conserved as well.

Therefore, the correct answer is c. both its momentum and its mechanical energy are conserved.

To determine the correct answer, let's analyze the physical principles involved.

When an object is dropped from rest with no air resistance, the only external force acting on it is gravity. According to Newton's second law of motion, the force acting on an object is equal to its mass multiplied by its acceleration. In this case, the acceleration is due to gravity, and it is constant for objects near the surface of the Earth.

Now let's consider the conservation principles:

1. Momentum Conservation:
Momentum is a property of moving objects and is defined as the product of an object's mass and its velocity. The law of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it.

2. Mechanical Energy Conservation:
Mechanical energy is the sum of an object's kinetic energy (energy due to motion) and potential energy (energy due to position). The law of conservation of mechanical energy states that the total mechanical energy of an isolated system remains constant when no external work or non-conservative forces are acting on it.

Based on the above principles, we can determine the correct answer:

a. only its momentum is conserved: This is incorrect because if no external forces act on the egg, both momentum and mechanical energy would be conserved.

b. only its kinetic energy is conserved: This is incorrect because potential energy (due to height) is also involved, and the total mechanical energy would be conserved.

c. both its momentum and its mechanical energy are conserved: This is correct. Since there are no external forces acting on the falling egg, both momentum (mv) and mechanical energy (KE + PE) are conserved.

d. its mechanical energy is conserved, but its momentum is not conserved: This is incorrect because both momentum and mechanical energy would be conserved.

Therefore, the correct answer is c. both its momentum and its mechanical energy are conserved.