Applied Maths - Mechanics 20

還不明白的薑

2009-07-31 3:31 am

A river of width a m with straight parallel banks flows due North with speed u ms^-1 . The points O and A are on opposite banks and A is due East of O. Coordinate axes Ox , Oy are taken in the East and North directions respectively. A boat, whose speed V ms^-1 relative to the water is constant, starts from O to A.

(a) If u is constant and equal to V/6 and the boat is steered so that it travels in a straight line towards A, find the time taken for the boat to travel from O to A.

(b) If u varies in such a way that u = x(a-x)V/a^2 and if the boat is steered due East, show that the coordinates (x,y) satisfies the differential equation

dy/dx = x(a-x)/a^2 .

If the boat reaches the East bank at C, calculate the distance AC and find the time taken.

回答 (1)

用戶9112

2009-07-31 5:08 am

✔ 最佳答案

(a)
The boat must be steered at an angle such that the resultant velocity is just perpendicular to the river. Without a diagram, it can be easily imagined that the y direction of the velocity is V/6 and the x direction velocity is U. Resultant velocity is just V so V2 = U2 + (V/6)2
U = (√35/6)V
Time taken = a/U = 6√35a/35V
(b)
u = dy/dt = x(a-x)V/a2 … (1)
V = dx/dt … (2)
Sub (2) into (1)
dy/dt = x(a-x)(dx/dt)/a2
dy/dx = x(a-x)/a2
dy = x(a-x)/a2 dx
Integrate both side,
y = (1/a2)(ax2/2 – x3/3) + K
Initial condition is x=y=0 so K = 0
when the boat reaches C, x = a
so y = (1/a2)(aa2/2 – a3/3) = a/6
AC = √(a2/62 + a2) = √37a/6
Time taken = a/V (using the x-direction distance and velocity)

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