8.13 Quick Revision -- Boats and Streams

Core Setup

    B = Speed of boat in still water
    S = Speed of stream / current

    ~~~~~~~~~~~~ current ~~~~~~~~~~~~>

    Downstream (with current):    Effective speed = B + S
    Upstream   (against current): Effective speed = B - S

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Finding B and S

    +---------------------------------------------+
    |  B = (Downstream + Upstream) / 2            |
    |  S = (Downstream - Upstream) / 2            |
    +---------------------------------------------+

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Time Formulas

    Time downstream = D / (B + S)
    Time upstream   = D / (B - S)

    Upstream always takes MORE time.

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Round Trip

    Total time = D/(B+S) + D/(B-S) = 2DB / (B^2 - S^2)

    Average speed = (B^2 - S^2) / B

    Alternative: Avg speed = 2 x (B+S) x (B-S) / [(B+S) + (B-S)]
                            (Harmonic mean of downstream and upstream speeds)

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n Times Longer Upstream

    If T_up = n x T_down:

    B / S = (n + 1) / (n - 1)

    n=2 --> B/S=3     n=3 --> B/S=2     n=5 --> B/S=3/2

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Two Boats on Same River

    +-----------------------------------------------------+
    |  STREAM SPEED CANCELS in relative speed!             |
    |                                                     |
    |  Same direction:  relative speed = |B1 - B2|        |
    |  Opposite:        relative speed = B1 + B2          |
    |                                                     |
    |  Use STILL WATER speeds only.                       |
    +-----------------------------------------------------+

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Floating Object (Hat / Ball Problem)

    +-----------------------------------------------------+
    |  Return time = Time spent going away                 |
    |  Stream speed is IRRELEVANT (reference frame trick)  |
    |                                                     |
    |  Total time floating = 2 x time going away           |
    |  Stream speed = distance floated / total time        |
    +-----------------------------------------------------+

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Ratio Shortcut

    T_up : T_down = p : q   (p > q)

    Upstream speed : Downstream speed = q : p  (inverted)

    B/S = (p+q) / (p-q)

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From Two Trip Equations

    Given same distance D, time downstream T1, upstream T2:

    B = D(T1+T2) / (2 x T1 x T2)
    S = D(T2-T1) / (2 x T1 x T2)

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Analogies

    Boats & Streams  =  Escalator + Person  =  Aircraft + Wind

    WITH the medium:   Effective = Own + Medium
    AGAINST the medium: Effective = Own - Medium

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Key Traps to Avoid

    1. "Speed in still water" is NOT downstream speed
       B ≠ B + S

    2. Average speed of round trip ≠ B
       It equals (B^2 - S^2)/B, which is LESS than B

    3. For two boats on same river, do NOT add stream speed
       to relative speed -- it cancels out

    4. If B <= S, the boat CANNOT go upstream

    5. Downstream time is always LESS than upstream time
       for the same distance

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Quick Problem-Solving Checklist

    [ ] Identify B and S (or downstream / upstream speeds)
    [ ] Determine direction: downstream or upstream?
    [ ] Apply correct formula: B+S or B-S
    [ ] For round trip: compute BOTH legs separately and add
    [ ] For two boats: remember stream cancels
    [ ] For floating object: use frame-of-reference trick
    [ ] Verify: upstream time > downstream time

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Part of Section 8.13: Boats and Streams