Nishtha on Fire!
Nishtha is standing on planet Brilliantia right now.. She fires a bullet vertically upwards with velocity v..
When the bullet reaches its maximum height, its acceleration due to the planet’s gravity is 1/4th of its value at the surface of the Brilliantia .
If the escape velocity from the Brilliantia is e = Nv , then the value of N is (you may ignore the energy loss due to Brilliantia's atmosphere)
The answer is 1.414213562.
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1 solution
Call The Mass Of The Body To Be m , Mass Of Planet Brilliantia To Be M ,Escape Velocity To Be E , Radius Of Brilliantia To Be R.
As Accleration due to gravity becomes 1/4th the height attained by body above the surface of brilliantia = R.
Using Conservation Of Energy
-GMm/R+1/2 m v^2= -GMm/2R
Solving We Get
v^2 = GM/R
Now For Escaping From Gravitational Pull Of The Planet Total Energy At The Surface Should Be Greater
Than Or Equal To Zero.
From There
-GMm/R+1/2 m E^2 =0
E^2 = 2GM/R
Comparing With Expression for v
We Get
E^2 = 2*v^2