This problem deals with the idea of the escape velocity of a...

This problem deals with the idea of the escape velocity of a particle from a body such as the Earth's surface. Recall from your course in physics that the potential energy of two masses, $m_{1}$ and $m_{2}$, separated by a distance $r$ is given by $$ V(r)=-\frac{G m_{1} m_{2}}{r} $$ (note the similarity with Coulomb's law) where $G=6.67 \times 10^{-11} \mathrm{~J} \cdot \mathrm{m} \cdot \mathrm{kg}^{-1}$ is called the gravitional constant. Suppose a particle of mass $m$ has a velocity $u$ perpendicular to the Earth's surface. Show that the minimum velocity that the particle must have in order to escape the Earth's surface (its escape velocity) is given by $$ u=\left(\frac{2 G M_{\text {earth }}}{R_{\text {earth }}}\right)^{1 / 2} $$ Given that $M_{\text {earth }}=5.98 \times 10^{24} \mathrm{~kg}$ is the mass of the Earth and $R_{\text {earth }}=6.36 \times 10^{6} \mathrm{~m}$ is its mean radius, calculate the escape velocity of a hydrogen molecule and a nitrogen molecule. What temperature would each of these molecules have to have so that their average speed exceeds their escape velocity?

This problem deals with the idea of the escape velocity of a particle from a body such as the Earth's surface. Recall from your course in physics that the potential energy of two masses, $m_{1}$ and $m_{2}$, separated by a distance $r$ is given by
$$
V(r)=-\frac{G m_{1} m_{2}}{r}
$$
(note the similarity with Coulomb's law) where $G=6.67 \times 10^{-11} \mathrm{~J} \cdot \mathrm{m} \cdot \mathrm{kg}^{-1}$ is called the gravitional constant. Suppose a particle of mass $m$ has a velocity $u$ perpendicular to the Earth's surface. Show that the minimum velocity that the particle must have in order to escape the Earth's surface (its escape velocity) is given by
$$
u=\left(\frac{2 G M_{\text {earth }}}{R_{\text {earth }}}\right)^{1 / 2}
$$
Given that $M_{\text {earth }}=5.98 \times 10^{24} \mathrm{~kg}$ is the mass of the Earth and $R_{\text {earth }}=6.36 \times 10^{6} \mathrm{~m}$ is its mean radius, calculate the escape velocity of a hydrogen molecule and a nitrogen molecule. What temperature would each of these molecules have to have so that their average speed exceeds their escape velocity?

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy associated with the position of a mass in a gravitational field. It is defined by the formula V(r) = -GMm/r, where G is the gravitational constant, M and m are the masses involved, and r is the separation distance. This concept is fundamental in understanding how objects interact under gravity and is used to calculate the energy required to move a mass within a gravitational field.

Escape Velocity

Escape velocity is the minimum speed an object must have to overcome the gravitational pull of a celestial body without further propulsion. It is derived by equating the object’s kinetic energy to the gravitational potential energy needed to break free from the gravitational influence, resulting in the formula u = (2GM/R)^(1/2), where M and R are the mass and radius of the celestial body, respectively.

Conservation of Energy

The conservation of energy principle states that energy in an isolated system cannot be created or destroyed but only transformed from one form to another. In the context of escape velocity, the kinetic energy imparted to a particle is converted into gravitational potential energy as it moves away from a gravitational source. This balance is critical in deriving and understanding the concept of escape velocity.

Kinetic Theory of Gases

The kinetic theory of gases explains how the microscopic motion of particles is related to the macroscopic observables such as temperature and pressure. It provides a link between the temperature of a gas and the average kinetic energy of its molecules, allowing us to relate molecular speeds to thermal energy. This connection is essential when determining the conditions under which a molecule’s speed exceeds a given threshold like the escape velocity.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the statistical distribution of speeds among particles in a gas, where the probability of a particle having a certain speed is related to the temperature of the gas. This distribution is used to understand and predict the fraction of molecules in a gas that can have speeds exceeding particular values, which is especially relevant when comparing molecular speeds to the escape velocity.

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