Question

A body is projected vertically upwards from the surface of earth with a velocity equal to half the escape velocity. If R is the radius of the earth, maximum height attained by the body from the surface of earth is

A R/6
B R/3
C 2R/3
D R/4

Answer 1

The maximum height attained by a body projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity is R/3.

Step-by-step explanation:

The maximum height attained by a body projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity can be determined using the concept of conservation of energy. The escape velocity is the minimum velocity required for the body to escape the gravitational pull of the earth. When the body reaches its maximum height, its velocity becomes zero. At this point, the total energy of the body is equal to zero. Using the formula for gravitational potential energy, we can equate the initial kinetic energy of the body to its potential energy at the maximum height.

Let's denote the escape velocity as v_esc, the initial velocity as v_0, and the maximum height as h. We can write the following equation:

0.5 * m * v_0^2 = m * g * R * (1 - 1/(1 + h/R))

where m is the mass of the body, g is the acceleration due to gravity, and R is the radius of the earth.

Since the initial velocity is half the escape velocity, we can substitute v_0 = 0.5 * v_esc into the equation:

0.5 * m * (0.5 * v_esc)^2 = m * g * R * (1 - 1/(1 + h/R))

By simplifying and solving for h, we can determine the maximum height attained:

h = R/3

Therefore, the correct answer is B) R/3.