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The kinetic energy needed to project a body of mass \(m\) from the earth surface (radius \(\mathrm{R}\)) to infinity is
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\(\mathrm{mgR} / 4\)
If any object through with escaping velocity then the object will never come back at the ground, So we can assume it will reach at infinite distance.
\(\mathrm{KE}=\frac{1}{2} \mathrm{mv} v_{\text {esc }}^2=\frac{1}{2} \mathrm{~m}(\sqrt{2 \mathrm{gR}})^2=\mathrm{mgR}\)
\(\mathrm{KE}=\frac{1}{2} \mathrm{mv} v_{\text {esc }}^2=\frac{1}{2} \mathrm{~m}(\sqrt{2 \mathrm{gR}})^2=\mathrm{mgR}\)
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