A body is projected vertically upwards from the surtace of a...

A body is projected vertically upwards from the surtace of a planet of radius $\mathrm{R}$ with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is $\ldots \ldots \ldots \ldots$ (A) $(\mathrm{R} / 3)$ (B) $(\mathrm{R} / 2)$ (C) $(\mathrm{R} / 4)$ (D) $(\mathrm{R} / 5)$

A body is projected vertically upwards from the surtace of a planet of radius $\mathrm{R}$ with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is $\ldots \ldots \ldots \ldots$
(A) $(\mathrm{R} / 3)$
(B) $(\mathrm{R} / 2)$
(C) $(\mathrm{R} / 4)$
(D) $(\mathrm{R} / 5)$

Key Concepts

Inverse Square Law

The gravitational force between two masses follows an inverse square law, meaning that the force (and consequently the gravitational potential energy) varies inversely with the square of the distance between the objects. This concept is crucial when analyzing gravitational effects over distances that are comparable to the size of the planet, as it affects how potential energy changes with altitude.

Gravitational Potential Energy

Gravitational potential energy in the context of a planetary body is defined by the formula U = -GMm/r, reflecting the work done against gravity to move a mass m from a distance r to infinity. Changes in gravitational potential energy during vertical motion are key to understanding how high a projectile can ascend before its kinetic energy is entirely expended.

Escape Velocity

Escape velocity is the minimum speed an object must have to break free from a planet's gravitational influence without further propulsion. It is determined by the relationship v_escape = ?(2GM/R) where G is the gravitational constant, M is the planet's mass, and R is its radius. In the context of the problem, the initial speed being a fraction of the escape velocity sets up a scenario for using energy analysis relative to the gravitational binding energy of the planet.

Conservation of Energy

This principle states that the total mechanical energy (the sum of kinetic and potential energy) of an isolated system remains constant. In problems involving vertical motion under gravity, energy is transformed between kinetic energy and gravitational potential energy. At the maximum height of the projectile, all the initial kinetic energy (adjusted for known gravitational parameters) has been converted into gravitational potential energy.

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