Given that mass of the earth is M and its radius is R. A...

Given that mass of the earth is $M$ and its radius is $R$. A body is dropped from a height equal to the radius of the earth above the surface of earth. When it reaches the ground its velocity will be (A) $\frac{G M}{R}$ (B) $\left[\frac{G M}{R}\right]^{1 / 2}$ (C) $\left[\frac{2 G M}{R}\right]^{1 / 2}$ (D) $\left[\frac{2 G M}{R}\right]$

Given that mass of the earth is $M$ and its radius is $R$. A body is dropped from a height equal to the radius of the earth above the surface of earth. When it reaches the ground its velocity will be
(A) $\frac{G M}{R}$
(B) $\left[\frac{G M}{R}\right]^{1 / 2}$
(C) $\left[\frac{2 G M}{R}\right]^{1 / 2}$
(D) $\left[\frac{2 G M}{R}\right]$

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Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy stored due to an object's position in a gravitational field. It is given by the expression -GMm/r for a mass m in the gravitational field of a body with mass M, reflecting how the energy changes with distance from the center of mass. This concept is crucial for calculating energy changes as an object moves within the field.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that in the absence of non-conservative forces, the total mechanical energy (the sum of kinetic and potential energies) remains constant. When an object moves in a gravitational field, the decrease in gravitational potential energy is converted into an increase in kinetic energy, allowing us to calculate speeds attained during free fall.

Newton's Law of Universal Gravitation

Newton's law of universal gravitation describes the force of attraction between two masses. It states that the gravitational force is proportional to the product of the masses and inversely proportional to the square of the distance between them. This foundational principle is used to derive expressions for gravitational potential energy and analyze the dynamics of objects in gravitational fields.

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