The river shown in the figure below flows north with a speed of 3 m/s. What shou
ID: 1587623 • Letter: T
Question
The river shown in the figure below flows north with a speed of 3 m/s. What should the minimum speed of a boat relative to the water to travel a straight line from C to D? What is the corresponding travel time? (Boat is going to travel North-East)
Given:
Width of river: 500m
Y axis length from Point C to Point D: 400m
Length of C to D therefore is the sqrt(500^2 + 400^2)
The answers are given as (Vboat/water) = (15/41)*sqrt(41) m/s ................. t = 820/3 s
I just need to know how to find those answers. Thanks!
Explanation / Answer
direction of travel from C to D
beta = tan^-1(400/500) = 38.66 degree
v_b/E = boat speed relative to earth
v_b/w = boat speed relative to water
v_b/E = v_b/E(icostheta + j sintheta) = v_b/w(icostheta + j sintheta)+ 3 j
tanbeta = (v_b/w * sin theta + 3 ) / v_b/w cos theta
v_b/w = (3*cosbeta)/(sin(beta - theta)
it will be minimum when dv_b/w/dtheta = 0
cos(beta-theta)(3osbeta)/sin^2(beta-theta)
it will be minimum when cos(beta-theta) = 0
beta-theta = pi/2
for minimum v_b/w = 3 cosbeta = 2.34 m/s
time = L/v
L = sqrt(500^2 + 400^2)
v = 2.34 m/s
t = 273.33 s
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