A balloon starts rising from the surface of the earth. The ascension rate is constant and equal to v₀. due to the wind the balloon gather the horizontal velocity component v _x=a y where is a constant and why is the height of ascent find the total tangential and normal acceleration of the balloon.?

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Solution

The path of the balloon will look something like this

After t sec ballon would have gone a height of v0tthen at that very instance the balloon's vx will be
vx=av0t⇒ax=av0
Vy=v0
ay=0
ar=totalacceleration=av0.......(1)
vtangentical is actually the resultant velocity
⇒vt=√v20+(av0t)2

⇒vt=v0√1+a2t2

$a subscript t equals fraction numerator d v subscript t over denominator d t end fraction equals fraction numerator v subscript 0 a squared t over denominator square root of 1 plus a squared t squared end root end fraction space space space space equals fraction numerator a squared y over denominator square root of 1 plus left parenthesis begin display style fraction numerator a y over denominator v subscript 0 end fraction end style right parenthesis squared end root end fraction N o w space w e space k n o w comma a subscript t squared plus a subscript N squared equals a subscript tau squared rightwards double arrow fraction numerator v subscript 0 squared a to the power of 4 t squared over denominator 1 plus a squared t squared end fraction plus a subscript N squared equals a squared v subscript 0 squared a subscript N equals square root of v subscript 0 squared a to the power of 4 t squared minus fraction numerator v subscript 0 squared a to the power of 4 t squared over denominator 1 plus a squared t squared end fraction end root space space space space space equals a v subscript 0 square root of fraction numerator 1 plus a squared t squared minus a squared t squared over denominator 1 plus a squared t squared end fraction end root space space space space space equals fraction numerator a v subscript 0 over denominator square root of 1 plus left parenthesis begin display style fraction numerator a y over denominator v subscript 0 end fraction end style right parenthesis squared end root end fraction$

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