Respuesta :
Answer:
a) The swimmer should travel perpendicular to the bank to minimize the spent in getting to the other side.
b) 133.33 m
c) 53.13°
d) 106.67 m
Explanation:
a) The swimmer should travel perpendicular to the bank to minimize the spent in getting to the other side.
b) velocity = distance * time
Let the velocity of the swimmer be [tex]v_{s}[/tex] = 1.5 m/s
The separation of the two sides of the river, d = 80 m
The time taken by the swimmer to get to the other end of the river bank,
[tex]t = \frac{d}{v_{s} }[/tex]
t = 80/1.5
t = 53.33 s
The swimmer will be carried downstream by the river through a distance, s
Let the velocity of the river be [tex]v_{r}[/tex] = 2.5 m/s
[tex]S = v_{r} t[/tex]
S = 53.33 * 2.5
S = 133.33 m
c) To minimize the distance traveled by the swimmer, his resultant velocity must be perpendicular to the velocity of the swimmer relative to water
That is ,
[tex]cos \theta = \frac{v_{s} }{v_{r} } \\cos \theta = 1.5/2.5\\cos \theta = 0.6\\\theta = cos^{-1} 0.6\\\theta = 53.13^{0}[/tex]
d) Downstream velocity of the swimmer, [tex]v_{y} = v_{s} sin \theta\\[/tex]
[tex]v_{y} = 1.5 sin 53.13\\v_{y} = 1.2 m/s[/tex]
The vertical displacement is given by, [tex]y = v_{y} t[/tex]
80 = 1.2 t
t = 80/1.2
t = 66.67 s
the horizontal speed,
[tex]v_{x} = 2.5 - 1.5cos53.13\\v_{x} = 1.6 m/s[/tex]
The downstream horizontal distance of the swimmer, [tex]x = v_{x} t[/tex]
x = 1.6 * 66.67
x = 106.67 m
a) To minimize time spent in water ; The swimmer should swim perpendicular to the banks
b) The swimmer will be carried 133.3 m downstream.
c) The direction swimmer should head in order to minimize the distance downstream is ; 53.13°
d) The swimmer will be carried 133.3 m downstream.
Given data:
velocity/speed of water ( [tex]v_{w}[/tex] ) = 2.5 m/s
distance between parallel banks = 80.0 m
speed of swimmer ( [tex]v_{s}[/tex] ) = 1.5 m/s
a) To minimize the time spent in water by the swimmer he/she should swim in a direction perpendicular to the banks.
b) Determine the downstream distance of the swimmer
first step ; determine time taken for swimmer to get the banks
t = d / [tex]v_{s}[/tex] = 80 m / 1.5 m/s = 53.33 secs
final step ; determine downstream distance that the swimmer is carried
s = [tex]v_{w} * t[/tex] = 2.5 * 53.33 = 133.3 m
c) Determine the direction to be headed in order to reduce downstream distance
we will apply the cosine rule
Cos ∅ = [tex]v_{s} / v_{w}[/tex]
= 1.5 / 2.5 = 0.6
cos ∅ = 0.6
∴ ∅ = 53.13°
Hence we can conclude that ; To minimize time spent in water, The swimmer should swim perpendicular to the banks,The swimmer will be carried 133.3 m downstream, The direction swimmer should head in order to minimize the distance downstream is ; 53.13°, The swimmer will be carried 133.3 m downstream.
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