A baloon starts raising from the surface of the earth. The ascent rate is constant and equal to V0. Due to thge wind the baloon geathers the horizontal velocity component Vx = ay, where  a is a constant and y is the height of ascent. Find how the following quantities depend on the height of ascent (a) the horizontal drift of the baloon and (b) the total tangential and normal acclerations of the baloon ?

Asked by sudhanshubhushanroy | 20th Jun, 2017, 01:11: PM

Expert Answer:

begin mathsize 12px style bold left parenthesis bold a bold right parenthesis space Given space that space
straight v subscript straight x space equals space ay space equals space dx over dt space......... space left parenthesis Equation space 1 right parenthesis
straight v subscript straight y space equals space straight v subscript 0 space equals space dy over dt space......... space left parenthesis Equation space 2 right parenthesis
From space the space above space equations comma space we space get
dy over dx space equals space straight v subscript 0 over ay space
or space ay over straight v subscript 0 space dy space equals space dx
rightwards arrow space integral subscript 0 superscript straight y ay over straight v subscript 0 space dy space equals integral subscript 0 superscript straight x space dx
or space straight x space equals space fraction numerator ay squared over denominator 2 straight v subscript 0 end fraction

bold left parenthesis bold b bold right parenthesis bold space space From space equation space 1 space and space 2 space we space have
space fraction numerator straight d squared straight x over denominator dt squared end fraction space equals space straight a dy over dt space equals space av subscript 0
space fraction numerator straight d squared straight y over denominator dt squared end fraction space equals space 0
Hence space we space have space
straight v space equals space ay space straight i space plus space straight v subscript 0 straight j
straight w space equals space av subscript 0 straight i space plus space 0 straight j
The space component space of space straight w space along space straight v space is space given space as
straight w space equals space fraction numerator straight w. straight v over denominator open vertical bar straight v close vertical bar end fraction space equals space fraction numerator straight a squared straight y straight v subscript 0 over denominator square root of straight a squared straight y squared space plus space straight v subscript 0 squared end root end fraction
The space component space of space straight w space normal space to space straight v space can space be space obtained space as
straight w subscript straight n space equals space square root of straight w squared minus straight w subscript straight i superscript 2 end root space equals space fraction numerator av subscript 0 superscript 2 over denominator square root of straight a squared straight y squared space plus space straight v subscript 0 squared end root end fraction
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Answered by Yashvanti Jain | 20th Jun, 2017, 03:24: PM