Question

A narrow tunnel is dug along the diameter of the earth, and a particle of massm, is placed at R/2 distance from the centre. Find the escape speed of theparticle from that place.​

Answer 1

The escape speed of the particle from that place V(min) = √2GM / R.

Explanation:

Correct statement:

A tunnel is dug along the diameter of the earth. There is particle of mass m at the centre of the tunnel. Find the minimum velocity given to the particle so that is just reaches to the surface of the earth. (R= radius of earth)

Solution:

Ei = P.E(i) + K.E (i)

Ei =  - 3/2 GMm/ R  + 1/2 x m x V(min)^2

E(f)  = p.E (f) + K.E (f)

       = - GMm / R

Using constant of energy.

E(i) =  E(f)

- 3/2 GMm/ R  + 1/2 x m x V(min)^2 = - GMm / R

1/2 m x v(min)^2 = 3/2 GMm /R - GMm / R

1/2 m x v(min)^2 = GMm /R

v(min)^2 = GMm / R

V(min) = √2GM / R

Thus the the escape speed of the particle from that place V(min) = √2GM / R.

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