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Question Number 18268 by Tinkutara last updated on 17/Jul/17
A balloon starts rising from the surface of earth. The ascension rate is constant and is equal to v_0 . Due to wind the balloon gathers horizontal velocity component v_x = ay, where a is a positive constant and y is the height of ascent. Find (i) The horizontal drift of the balloon x(y), (ii) The total, tangential and normal accelerations of the balloon.
$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{starts}\:\mathrm{rising}\:\mathrm{from}\:\mathrm{the}\:\mathrm{surface} \\ $$$$\mathrm{of}\:\mathrm{earth}.\:\mathrm{The}\:\mathrm{ascension}\:\mathrm{rate}\:\mathrm{is}\:\mathrm{constant} \\ $$$$\mathrm{and}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:{v}_{\mathrm{0}} .\:\mathrm{Due}\:\mathrm{to}\:\mathrm{wind}\:\mathrm{the} \\ $$$$\mathrm{balloon}\:\mathrm{gathers}\:\mathrm{horizontal}\:\mathrm{velocity} \\ $$$$\mathrm{component}\:{v}_{{x}} \:=\:{ay},\:\mathrm{where}\:{a}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive} \\ $$$$\mathrm{constant}\:\mathrm{and}\:{y}\:\mathrm{is}\:\mathrm{the}\:\mathrm{height}\:\mathrm{of}\:\mathrm{ascent}. \\ $$$$\mathrm{Find} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{The}\:\mathrm{horizontal}\:\mathrm{drift}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon} \\ $$$${x}\left({y}\right), \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{The}\:\mathrm{total},\:\mathrm{tangential}\:\mathrm{and}\:\mathrm{normal} \\ $$$$\mathrm{accelerations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon}. \\ $$
Answered by ajfour last updated on 16/Aug/17
Commented by Tinkutara last updated on 17/Aug/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by ajfour last updated on 16/Aug/17
y=v_0 t , v_x =ay=av_0 t so ∫_0 ^( x) dx =∫_0 ^( t) av_0 tdt =((av_0 t^2 )/2) ⇒ x=((ay^2 )/(2v_0 )) . a_y =0 ; a_(net) ^� =((dv_x /dt))i^� =av_0 i^� . a_t =∣a_(nrt) ^� ∣cos θ =av_0 (v_x /( (√(v_x ^2 +v_y ^2 )))) a_t =((av_0 (av_0 t))/( (√((av_0 t)^2 +v_0 ^2 )))) =((av_0 t)/( (√(a^2 t^2 +1)))) . a_N =∣a_(net) ∣sin θ =av_0 (v_y /( (√(v_x ^2 +v_y ^2 )))) a_N =((av_0 )/( (√(a^2 t^2 +1)))) .
$$\:\:{y}={v}_{\mathrm{0}} {t}\:\:,\:\:{v}_{{x}} ={ay}={av}_{\mathrm{0}} {t} \\ $$$$\:{so}\:\:\:\:\int_{\mathrm{0}} ^{\:\:{x}} {dx}\:=\int_{\mathrm{0}} ^{\:\:{t}} {av}_{\mathrm{0}} {tdt}\:=\frac{{av}_{\mathrm{0}} {t}^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\:\:\:\:\Rightarrow\:\:\:\:\:{x}=\frac{{ay}^{\mathrm{2}} }{\mathrm{2}{v}_{\mathrm{0}} }\:\:\:. \\ $$$$\:\:\:\:\:\:\:{a}_{{y}} =\mathrm{0}\:;\:\:\:\:\bar {\boldsymbol{{a}}}_{\boldsymbol{{net}}} =\left(\frac{{dv}_{{x}} }{{dt}}\right)\hat {{i}}\:={av}_{\mathrm{0}} \hat {{i}}\:. \\ $$$$\:\:\:\:\:\:\boldsymbol{{a}}_{\boldsymbol{{t}}} =\mid\bar {{a}}_{{nrt}} \mid\mathrm{cos}\:\theta\:\:={av}_{\mathrm{0}} \frac{{v}_{{x}} }{\:\sqrt{{v}_{{x}} ^{\mathrm{2}} +{v}_{{y}} ^{\mathrm{2}} }}\: \\ $$$$\:\:\boldsymbol{{a}}_{\boldsymbol{{t}}} \:=\frac{\boldsymbol{{av}}_{\mathrm{0}} \left(\boldsymbol{{av}}_{\mathrm{0}} \boldsymbol{{t}}\right)}{\:\sqrt{\left(\boldsymbol{{av}}_{\mathrm{0}} \boldsymbol{{t}}\right)^{\mathrm{2}} +\boldsymbol{{v}}_{\mathrm{0}} ^{\mathrm{2}} }}\:=\frac{\boldsymbol{{av}}_{\mathrm{0}} \boldsymbol{{t}}}{\:\sqrt{\boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{t}}^{\mathrm{2}} +\mathrm{1}}}\:. \\ $$$$\:\:\boldsymbol{{a}}_{\boldsymbol{{N}}} =\mid{a}_{{net}} \mid\mathrm{sin}\:\theta\:={av}_{\mathrm{0}} \frac{{v}_{{y}} }{\:\sqrt{{v}_{{x}} ^{\mathrm{2}} +{v}_{{y}} ^{\mathrm{2}} }}\: \\ $$$$\:\:\boldsymbol{{a}}_{\boldsymbol{{N}}} =\frac{\boldsymbol{{av}}_{\mathrm{0}} }{\:\sqrt{\boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{t}}^{\mathrm{2}} +\mathrm{1}}}\:\:. \\ $$

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