Respuesta :
Answer:
1) [tex]\Delta s=1000\ ft[/tex]
2) [tex]\Delta s'=998.11\ ft.s^{-1}[/tex]
3) [tex]t\approx125\ s[/tex]
[tex]t'\approx463.733\ s[/tex]
Explanation:
Given:
width of river, [tex]w=500\ ft[/tex]
speed of stream with respect to the ground, [tex]v_s=8\ ft.s^{-1}[/tex]
speed of the swimmer with respect to water, [tex]v=4\ ft.s^{-1}[/tex]
Now the resultant of the two velocities perpendicular to each other:
[tex]v_r=\sqrt{v^2+v_s^2}[/tex]
[tex]v_r=\sqrt{4^2+8^2}[/tex]
[tex]v_r=8.9442\ ft.s^{-1}[/tex]
Now the angle of the resultant velocity form the vertical:
[tex]\tan\beta=\frac{v_s}{v}[/tex]
[tex]\tan\beta=\frac{8}{4}[/tex]
[tex]\beta=63.43^{\circ}[/tex]
- Now the distance swam by the swimmer in this direction be d.
so,
[tex]d.\cos\beta=w[/tex]
[tex]d\times \cos\ 63.43=500[/tex]
[tex]d=1118.034\ ft[/tex]
Now the distance swept downward:
[tex]\Delta s=\sqrt{d^2-w^2}[/tex]
[tex]\Delta s=\sqrt{1118.034^2-500^2}[/tex]
[tex]\Delta s=1000\ ft[/tex]
2)
On swimming 37° upstream:
The velocity component of stream cancelled by the swimmer:
[tex]v'=v.\cos37[/tex]
[tex]v'=4\times \cos37[/tex]
[tex]v'=3.1945\ ft.s^{-1}[/tex]
Now the net effective speed of stream sweeping the swimmer:
[tex]v_n=v_s-v'[/tex]
[tex]v_n=8-3.1945[/tex]
[tex]v_n=4.8055\ ft.s^{-1}[/tex]
The component of swimmer's velocity heading directly towards the opposite bank:
[tex]v'_r=v.\sin37[/tex]
[tex]v'_r=4\sin37[/tex]
[tex]v'_r=2.4073\ ft.s^{-1}[/tex]
Now the angle of the resultant velocity of the swimmer from the normal to the stream:
[tex]\tan\phi=\frac{v_n}{v'_r}[/tex]
[tex]\tan\phi=\frac{4.8055}{2.4073}[/tex]
[tex]\phi=63.39^{\circ}[/tex]
- Now let the distance swam in this direction be d'.
[tex]d'\times \cos\phi=w[/tex]
[tex]d'=\frac{500}{\cos63.39}[/tex]
[tex]d'=1116.344\ ft[/tex]
Now the distance swept downstream:
[tex]\Delta s'=\sqrt{d'^2-w^2}[/tex]
[tex]\Delta s'=\sqrt{1116.344^2-500^2}[/tex]
[tex]\Delta s'=998.11\ ft.s^{-1}[/tex]
3)
Time taken in crossing the rive in case 1:
[tex]t=\frac{d}{v_r}[/tex]
[tex]t=\frac{1118.034}{8.9442}[/tex]
[tex]t\approx125\ s[/tex]
Time taken in crossing the rive in case 2:
[tex]t'=\frac{d'}{v'_r}[/tex]
[tex]t'=\frac{1116.344}{2.4073}[/tex]
[tex]t'\approx463.733\ s[/tex]
