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Sheena can row a boat at 3.13 mi/h in still water. She needs to cross a river that is 1.10 mi wide with a current flowing at 1.75 mi/h. Not having her calculator ready, she guesses that to go straight across, she should head 60.0° upstream.
(a) What is her speed with respect to the starting point on the bank?
(b) How long does it take her to cross the river?
(c) How far upstream or downstream from her starting point will she reach the opposite bank?
(d) In order to go straight across, what angle upstream should she have headed?

  • physics -

    Let u : her speed across the stream
    Let v : her speed up the stream
    Find w : her speed relative to the bank

    u = 3.13 cos(60°) [mi/h]
    v = 3.13 sin(60°) - 1.35 [mi/h]

    Her speed relative to the bank.

    w = √(u^2+v^2)
    w = √((3.13^2+(3.13√3-1.35)^2)/2) [mi/h]

    Time to cross the stream:
    Let x : the distance across the bank
    Find t : the time to cross the stream

    x = 1.10[mi]

    t = x/u
    t = 1.10/(3.13 cos(60°)) [hr]

    Distance traveled up
    Find y : the distance upstream

    y = vt
    y = 1.10(3.13 sin(60°)-1.35)/(3.13 cos(60°)) [mi]

    Let θ : angle needed to travel straight across.
    Ergo: to have a zero upstream velocity.

    0 = 3.13 sin(θ) - 1.35
    θ = arcsin(1.35/3.13)

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