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Step-by-Step Solution
Step 1: Understand the Problem
We want to find the initial velocity v_i needed to project a body vertically upwards from the Earth’s surface so that it reaches a height of 10R (where R is Earth’s radius). This velocity v_i is related to the escape velocity v_e by an expression:
v_i = \sqrt{\frac{x}{y}} \times v_e.
We need to determine the value of x in the fraction \sqrt{\frac{x}{y}} .
Step 2: Write Down Relevant Equations
1. Gravitational potential energy of a mass m at a distance r from Earth’s center (assuming Earth has mass M_e ) is
U = -\frac{G\,M_e\,m}{r} ,
where G is the gravitational constant.
2. The escape velocity v_e from the Earth’s surface is given by
v_e = \sqrt{\frac{2G\,M_e}{R}}.
Step 3: Apply Energy Conservation
When the body is launched from Earth’s surface with initial velocity v_i , its total energy (potential + kinetic) at the surface must equal its total energy at the maximum height (where its final kinetic energy is zero):
Initial total energy at Earth’s surface:
E_\text{initial} = \underbrace{\left(- \frac{G\,M_e\,m}{R}\right)}_{\text{potential energy}} + \underbrace{\frac{1}{2}m\,v_i^2}_{\text{kinetic energy}}.
Final total energy at r = R + 10R = 11R (the maximum height):
E_\text{final} = -\frac{G\,M_e\,m}{11R} + 0 \quad (\text{since final kinetic energy }=0).
By conservation of energy,
-\frac{G\,M_e\,m}{R} + \frac{1}{2}m\,v_i^2 = -\frac{G\,M_e\,m}{11R}.
Step 4: Solve for v_i
Rearrange the energy equation:
\frac{1}{2} m\,v_i^2 = -\frac{G\,M_e\,m}{11R} + \frac{G\,M_e\,m}{R} = \frac{10\,G\,M_e\,m}{11R}.
Canceling m on both sides and multiplying by 2,
v_i^2 = \frac{20\,G\,M_e}{11R}.
Hence,
v_i = \sqrt{\frac{20\,G\,M_e}{11R}}.
Step 5: Express v_i in Terms of v_e
Using the expression for escape velocity v_e = \sqrt{\frac{2\,G\,M_e}{R}} , we rewrite v_i :
v_i = \sqrt{\frac{20\,G\,M_e}{11R}}
= \sqrt{\frac{20}{11} \times \frac{G\,M_e}{R}}
= \sqrt{\frac{10}{11}} \times \sqrt{\frac{2\,G\,M_e}{R}}
= \sqrt{\frac{10}{11}} \, v_e.
In the form v_i = \sqrt{\frac{x}{y}}\;v_e , we see
\sqrt{\frac{x}{y}} = \sqrt{\frac{10}{11}}.
So, x = 10 .
Answer
The value of x is 10.
