Question #185083

A man falling vertivally by parachute in a steady downpour of rain observes that when his speed is V1 the rain appears to make angle α with the upward vertical.When his speed is V2(V2>V1) the rain appears to make angle ß with the upward vertical.

Show that the rain actually falls at an angle Θ with the vertical given by

(V2-V1)cotΘ=V2cotα-V1cotß.


1
Expert's answer
2021-05-07T09:20:02-0400

The velocity of man is V1=V1j.\vec V_{1}=-V_1\vec j.

Suppose the relative velocity of the wind is


VWM1=asin(α)i+acos(α)j\vec V_{WM_1}=a\sin(\alpha)\vec i+a\cos(\alpha)\vec j

The velocity of the wind is


VW=V1+VWM1=asin(α)i+(V1+acos(α))j\vec V_W=\vec V_{1}+\vec V_{WM_1}=a\sin(\alpha)\vec i+(-V_1+a\cos(\alpha))\vec j


The velocity of man is V2=V2j.\vec V_{2}=-V_2\vec j.

Suppose the relative velocity of the wind is


VWC2=bsin(β)i+bcos(β)j\vec V_{WC2}=b\sin(\beta)\vec i+b\cos(\beta)\vec j


The velocity of the wind is


VW=V2+VWC2=bsin(β)i+(V2+bcos(β))j\vec V_W=\vec V_{2}+\vec V_{WC_2}=b\sin(\beta)\vec i+(-V_2+b\cos(\beta))\vec j


Then


asin(α)=bsin(β)V1+acos(α)=V2+bcos(β)\begin{matrix} a\sin(\alpha) =b\sin(\beta) \\ -V_1+a\cos(\alpha)=-V_2+b\cos(\beta) \end{matrix}


a=bsin(β)sin(α)V2V1=bsin(β)(cot(β)cot(α))\begin{matrix} a =b\cdot\dfrac{\sin(\beta)}{\sin(\alpha)} \\ V_2-V_1=b\sin(\beta)(\cot(\beta)-\cot(\alpha)) \end{matrix}

cot(Θ)=V2+bcos(α)bsin(β)\cot(\Theta)=\dfrac{-V_2+b\cos(\alpha)}{b\sin(\beta)}

bsin(β)cot(Θ)=V2+bsin(β)cot(β)b\sin(\beta)\cot(\Theta)=-V_2+b\sin(\beta)\cot(\beta)

(V2V1)cot(Θ)=V2((cot(β)cot(α)))(V_2-V_1)\cot(\Theta)=-V_2((\cot(\beta)-\cot(\alpha)))

+(V2V1)cot(β)+(V_2-V_1)\cot(\beta)

(V2V1)cot(Θ)=V2cot(α)V1cot(β)(V_2-V_1)\cot(\Theta)=V_2\cot(\alpha)-V_1\cot(\beta)



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