4.11 Projection in gravitational field  (Page 4/5)

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$M=5.98X{10}^{24}\phantom{\rule{1em}{0ex}}kg$

$R=6.37X{10}^{6}\phantom{\rule{1em}{0ex}}m$

$⇒{v}_{e}=\sqrt{\left(\frac{2X6.67X{10}^{-11}X5.98X{10}^{24}}{6.37X{10}^{6}}\right)}$

$⇒{v}_{e}=11.2\phantom{\rule{1em}{0ex}}km/s$

We should understand that this small numerical value is deceptive. Actually, it is almost impossible to impart such magnitude of speed. Let us have a look at the magnitude in terms of “km/hr”,

${v}_{e}=11.2\phantom{\rule{1em}{0ex}}km/s=11.2X60x60=40320\phantom{\rule{1em}{0ex}}km/hr$

If we compare this value with the speed of modern jet (which moves at 1000 km/hr), this value is nearly 40 times! It would be generally a good idea to project object from an artificial satellite instead, which itself may move at great speed of the order of about 8-9 km/s. The projectile would need only additional 2 or 3 km/hr of speed to escape, if projected in the tangential direction of the motion of the satellite.

This mechanism is actually in operation in multistage rockets. Each stage acquires the speed of previous stage. The object (probe or vehicle) can, then, be let move on its own in the final stage to escape Earth’s gravity. The mechanism as outlined here is actually the manner an interstellar probe or vehicle is sent out of the Earth’s gravitational field. We can also appreciate that projection, in this manner, has better chance to negotiate friction effectively as air resistance at higher altitudes is significantly less or almost negligible.

This discussion of escape velocity also underlines that the concept of escape velocity is related to object, which is not propelled by any mechanical device. An object, if propelled, can escape gravitational field at any speed.

Escape velocity of Moon :

In the case of Earth’s moon,

$M=7.4X{10}^{20}\phantom{\rule{1em}{0ex}}kg$

$R=1.74X{10}^{6}\phantom{\rule{1em}{0ex}}m$

$⇒{v}_{e}=\sqrt{\left(\frac{2X6.67X{10}^{-11}X7.4X{10}^{20}}{1.74X{10}^{6}}\right)}$

$⇒{v}_{e}=2.4\phantom{\rule{1em}{0ex}}km/s$

The root mean square velocity of gas is greater than this value. This is the reason, our moon has no atmosphere. Since sound requires a medium to propagate, we are unable to talk directly there as a consequence of the absence of atmosphere.

Direction of projection

It may appear that we may need to fire projectile vertically to let it escape in interstellar space. This is not so. The spherical symmetry of Earth indicates that we can project body in any direction with the velocity as determined such that it clears physical obstructions in its path. From this point of view, the term “velocity” is a misnomer as direction of motion is not involved. It would have been more appropriate to call it “speed”.

The direction, however, makes a difference in escape velocity for some other reason. The Earth rotates in particular direction – it rotates from East to west at a linear speed of 465 m/s. So if we project the body in the tangential direction east-ward, then Earth rotation helps body’s escape. The effective escape velocity is 11200 – 465 = 10735 m/s. On the other hand, if we project west-ward, then escape velocity is 11200 + 465 = 11635 m/s.

Escape velocity and black hole

Black holes are extremely high density mass. This represents the final stage of evolution of a massive star, which collapses due to its own gravitational force. Since mass remains to be very large while radius is reduced (in few kms), the gravitational force becomes extremely large. Such great is the gravitational force that it does not even allow light to escape.

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