Escape velocity
Escape velocity :
We will learn about binding energy which is the minimum energy is given to the particle in any form (normally kinetic), the particle no longer remains attached to the earth. It goes out of the gravitational field of earth.
We will further learn that about Escape velocity which is the minimum velocity to be imparted to a body from the surface of the earth (or planet) such that it just escapes the gravitational pull of the earth (i.e. reaches infinity and stops there) is called the escape velocity.
When a body is launched from the surface of the earth with the minimum speed, say escape speed ve then it will reach infinity with zero speed.
Binding Energy
Total mechanical energy (potential + kinetic) of a closed system is negative. The modulus of this total mechanical energy is known as the binding energy of the system. This is the energy due to which system is closed or different parts of the system are bound to each other. Suppose the mass m is placed on the surface of earth.
The radius of the earth is R and its mass is M. Then, the kinetic energy of the particle K = 0 and the potential energy is U = –
Therefore, the total mechanical energy is,
E = K + U = 0 –
E = –
Binding energy =
It is due to this energy, the particle is attached with the earth. If minimum this much energy is given to
the particle in any form (normally kinetic) the particle no longer remains attached to the earth. It goes
out of the gravitational field of earth.
Escape Velocity
Escape Velocity :
When a stone is thrown up it goes up to a maximum height and then returns. As the particle goes up, the gravitational potential energy increases and the kinetic energy of the particle decreases. The particle will continue to go up till its kinetic energy becomes zero and will return from there.
The minimum velocity to be imparted to a body from the surface of the earth (or planet) such that it just escapes the gravitational pull of the earth (i.e. reaches infinity and stops there) is called the escape velocity. When a body is launched from the surface of the earth with the minimum speed, say escape speed ve then it will reach infinity with zero speed.
Now consider the binding energy of a particle on the surface of earth kept at rest is
If
this much energy in the form of kinetic energy is supplied to the particle, it leaves the gravitational
field of the earth. So, if ve is the escape velocity of the particle, then.
or
or
as
Substituting the value of g (9.8 m/s
2
) and R (6.4 × 10
6
m), we get
v
e
= 11.2 Km/s
This critical initial velocity is called the escape velocity. Putting the values of G, M and R, the escape velocity from the earth comes out to be 11. 6 km s
–1
. In this we have neglected the effect of other planets, stars and other objects in space. In fact, even if the initial velocity is somewhat less than the escape velocity,
the particle may get attracted by some other celestial object and land up there Thus, the minimum velocity needed to take a particle to infinity from the earth is called the escape velocity. On the surface of earth its value is 11.2 km/s. Note that the escape speed does not depend on the mass of the body launched from the surface of the earth. Also, escape velocity is independent of the angle of launch or the angle of projection.
Important Points To Remember
(a)
If the particle is launched from the surface of the earth with a velocity u >v
e
, then the body will reach infinity with a non zero velocity v which can be calculated by using the Law of Conservation of Energy, so
(U + K)
at surface
= (U + K)
at ∞
⇒
(b)
If the particle is launched from the surface of the earth with a velocity u v < e , then the body will attain a maximum height h (say) and then return to the surface of the earth. The maximum height h can be calculated by using the Law of Conservation of Energy, so
(U + K)
at surface
= (U + K)
at h
⇒
(c)
Escape speed depends on the mass and size of the planet. That is why escape velocity on the Jupiter is more than that on the earth.
(d)
Escape speed is independent of the mass of the body.
(e)
Any body thrown upward with escape speed starts moving around the Sun.
i.
The value of escape speed of a body does not depend upon the mass (m) of the body and its angle of projection from the surface of earth or any other planet.
ii.
The value of escape speed depends upon the mass and radius of the planet from the surface of which the body is to be projected. Clearly, the value of escape speed of a body will be different for different planets.
