The escape speed is the minimum speed with which a body must be projected in order that it may escape from the gravitational pull of the planet.

**Escape speed**

If we throw a body upwards, it
reaches a certain height and then falls back. This is due to the gravitational
attraction of the Earth. If we throw the body with a greater speed, it rises to
a greater height. If the body is projected with a speed of 11.2 km/s, it
escapes from the Earth and never comes back. *The escape speed is the minimum speed with* *which a body must be projected in order that it may escape from the
gravitational pull of the planet.*

Consider a body of mass m placed
on the Earth's surface. The gravitational potential energy is E_{P} =
-GMm /R

where M is the mass of the Earth
and R is its radius.

If the body is projected up with
a speed v_{e}, the kinetic energy is

E_{K} = ½ mv_{e}^{2}

the initial total energy of the
body is

E_{i} = ½ mv_{e}^{2} - GMm/R ……………………..(1)

If the body reaches a height h
above the Earth's surface, the gravitational potential energy is

E_{P} = - GMm / ( R+h)

Let the speed of the body at the
height is v, then its kinetic energy is,

E_{p} = ½ mv^{2}

Hence, the final total energy of
the body at the height is

E_{f} = ½ mv^{2}
- GMm(R+h) ………… (2)

We know that the gravitational
force is a conservative force and hence the total mechanical energy must be
conserved.

E_{i} = E_{f}

The body will escape from the
Earth's gravity at a height where the gravitational field ceases out. (i.e) h =
∞. At the height h = ∞, the speed v of the body is zero.

[Mv_{e}^{2} /
2 ] - [GMm/R] = 0

From the relation g = GM/R^{2}

we get GM = gR^{2}

Thus, the escape speed is v_{e}
= root(2gR)

The escape speed for Earth is
11.2 km/s, for the planet Mercury it is 4 km/s and for Jupiter it is 60 km/s.
The escape speed for the moon is about 2.5 km/s.

*An interesting consequence of
escape speed with the atmosphere of a planet*

We know that the escape speed is
independent of the mass of the body. Thus, molecules of a gas and very massive
rockets will require the same initial speed to escape from the Earth or any
other planet or moon.

The molecules of a gas move with
certain average velocity, which depends on the nature and temperature of the
gas. At moderate temperatures, the average velocity of oxygen, nitrogen and
carbon-di-oxide is in the order of 0.5 km/s to 1 km/s and for lighter gases
hydrogen and helium it is in the order of 2 to 3 km/s. It is clear that the
lighter gases whose average velocities are in the order of the escape speed,
will escape from the moon. The gravitational pull of the moon is too weak to
hold these gases. The presence of lighter gases in the atmosphere of the Sun
should not surprise us, since the gravitational attraction of the sun is very
much stronger and the escape speed is very high about 620 km/s.

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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Escape speed derivation and with the atmosphere of a planet |

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