Question 248: Gravitation

Question

The initial velocity v
i
required to project a body vertically upward from the surface of the earth to reach a height of 10R, where R is the radius of the earth, may be described in terms of escape velocity v
e
such that $${v_i} = \sqrt {{x \over y}}  	imes {v_e}$$. The value of x will be ____________.

Accepted Answer

Here R = radius of the earth
From energy conservation
$${{ - G{m_e}m} \over R} + {1 \over 2}m{v_i}^2 = {{ - G{m_e}m} \over {11R}} + 0$$
$${1 \over 2}m{v_i}^2 = {{10} \over {11}}{{G{m_e}m} \over R}$$
$${v_i} = \sqrt {{{20} \over {11}}{{G{m_e}} \over R}} $$
$${v_i} = \sqrt {{{10} \over {11}}} {v_e}$$
{$$ \because $$ escape velocity $${v_e} = \sqrt {{{2G{m_e}} \over R}} $$}
Then the value of x = 10

Gravitation

Integer / Numeric Answer Type