54.6: Escape Velocity
- Page ID
- 92345
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The escape velocity is the initial velocity a particle must have to escape the gravity of its parent body. Typically it refers to the initial velocity a particle must have at the surface of a planet in order to leave the planet forever, and never be pulled back by the planet's gravity. If a particle leaves the surface of a planet with an initial velocity equal to the escape velocity, then the body will move more and more slowly as the particle moves farther from the planet, finally reaching a velocity of zero at \(r=\infty\). (We assume only the particle and the planet are present, and ignore all other bodies.)
To compute the escape velocity, consider running the problem with time running backwards: the body starts at \(r=\infty\) with zero velocity and falls toward the planet. The impact velocity from infinity will be the same as the escape velocity. Now at \(r=\infty\), the potential energy \(U=-G M_{p} m / r=0\), where \(M_{p}\) is the mass of the planet and \(m\) is the mass of the particle. Since the particle is at rest at \(r=\infty\), the kinetic energy there is also zero, so the total mechanical energy \(K+U=0\). Now let the particle begin falling from \(r=\infty\) under the influence of the planet's gravity, until it impacts the planet at \(r=R_{p}\), where \(R_{p}\) is the radius of the planet. At the point of impact the potential energy is \(U=-G M_{p} m / R_{p}\), and its kinetic energy will be \(K=m v_{e}^{2} / 2\), where \(v_{e}\) is the impact (escape) velocity. By the law of conservation of energy, the total mechanical energy at \(r=\infty\) must be the same as it is at \(r=R_{p}\) :
\[K+U=\frac{1}{2} m v_{e}^{2}-\frac{G M_{p} m}{R_{p}}=0\]
Solving for the escape velocity \(v_{\boldsymbol{e}}\), we find
\[v_{e}=\sqrt{\frac{2 G M_{p}}{R_{p}}}\]
For the Earth, for example, we have (from Appendix L) \(G M_{p}=3.986005 \times 10^{14} \mathrm{~m}^{3} \mathrm{~s}^{-2}\) and \(R_{p}=\) \(6378.140 \times 10^{3} \mathrm{~m}\); substituting these values into Eq. \(\PageIndex{1}\), we find the escape velocity for Earth is \(v_{e}=11.2\) \(\mathrm{km} / \mathrm{s}\). In other words, if you were to fire a projectile from the surface of the Earth with an initial velocity of \(11.2 \mathrm{~km} / \mathrm{s}\), it would be able to escape the Earth's gravity, going more and more slowly the higher it goes, finally coming to rest at \(r=\infty\).