Calculating the escape velocity from earth
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The following quantities appear in the problem:
Masse \(m\) / Kraft \(F\) / Energie \(E\) / Geschwindigkeit \(v\) / Radius \(r\) /
The following formulas must be used to solve the exercise:
\(F = G \dfrac{m_1m_2}{r^2} \quad \) \(E_{\rm \scriptscriptstyle kin} = \dfrac12 mv^2 \quad \)
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Exercise:
What is the minimum speed required for an object to be launched vertically from the Earth's surface in order to escape the Earth's gravitational field? The Earth's mass is MO and its radius RO.
Solution:
The work required to move an object from the Earth's surface to infinity is: W _R^infty Fr ddr _R^infty G fracm Mr^ ddr G m M _R^infty fracr^ ddr G m M left -fracr right_R^infty G m M left + fracR right fracG m MR The object must initially be given so much kinetic energy which is why the minimum speed at launch is calculated as follows: sscEkin &mustbe W frac m v^ fracG m MR v^ frac G MR v sqrtfrac G MR sqrtfrac ncG MR v
What is the minimum speed required for an object to be launched vertically from the Earth's surface in order to escape the Earth's gravitational field? The Earth's mass is MO and its radius RO.
Solution:
The work required to move an object from the Earth's surface to infinity is: W _R^infty Fr ddr _R^infty G fracm Mr^ ddr G m M _R^infty fracr^ ddr G m M left -fracr right_R^infty G m M left + fracR right fracG m MR The object must initially be given so much kinetic energy which is why the minimum speed at launch is calculated as follows: sscEkin &mustbe W frac m v^ fracG m MR v^ frac G MR v sqrtfrac G MR sqrtfrac ncG MR v