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An object is projected from earth's surface with escape velocity at 30°with horizontal . What is the angle made by velocity with horizontal when object reaches height 2R from earth's surface?

Asked by jainkiran2014 6th January 2018, 9:54 AM
Answered by Expert
Answer:
This problem can be solved by using the following conservation of energy relation in gravitational field.
 
begin mathsize 12px style 1 half m v squared space minus space fraction numerator G M m over denominator r end fraction space equals space c o n s tan t end style
m - mass of particle, M - mass of earth, v - velocity of particle, G - gravitational constant and r is position of particle from center of earth.
 
Initially object is projected at earth's surface with escape velocity.  It is required to get the velocity at a height of 2R (R is earth's radius) above earths surface.  Let us equate the total energy at both the positions as follows.
 
begin mathsize 12px style 1 half m v subscript i superscript 2 space minus space fraction numerator G M m over denominator R end fraction space equals fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction m v subscript f superscript 2 space minus space fraction numerator begin display style G M m end style over denominator begin display style 3 R end style end fraction space space v subscript f superscript 2 space equals space v subscript i superscript 2 space minus space 4 over 3 fraction numerator G M m over denominator R end fraction sin c e space v subscript i space equals space e s c a p e space v e l o c i t y space space v subscript e equals space square root of fraction numerator 2 G M over denominator R end fraction end root comma space w e space w i l l space g e t space v subscript f superscript 2 space equals fraction numerator v subscript e superscript 2 over denominator 3 end fraction h e n c e space v subscript f space equals fraction numerator v subscript e over denominator square root of 3 end fraction end style
Let α be the angle that velocity vf makes with horizontal. It should be noted here initially the object is projected at an angle 30º at earths surface and the horizontal component of this velocity is unaffected. 
then we can write the component form of the velocity as given below
begin mathsize 12px style fraction numerator v subscript e superscript 2 over denominator 3 end fraction sin squared alpha space plus space v subscript e superscript 2 cos squared space 30 space equals v subscript e superscript 2 comma  s o l v i n g space f o r space alpha comma space w e space g e t space alpha space equals space pi over 3 end style
Answered by Expert 7th January 2018, 7:55 PM
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Tags: gravitation
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