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Boats & Streams

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Quick recall

  • Downstream adds the current; upstream subtracts it.
  • The boat must be faster than the current to travel upstream.

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Formula Cards

1. Downstream speed

vd=u+vv_d=u+v

u = boat speed in still water; v = stream speed.

2. Upstream speed

vu=u−vv_u=u-v

Requires u > v for upstream travel.

3. Recover boat speed

u=vd+vu2u=\frac{v_d+v_u}{2}

4. Recover stream speed

v=vd−vu2v=\frac{v_d-v_u}{2}

5. Average round-trip speed

vˉ=2vdvuvd+vu\bar{v}=\frac{2v_dv_u}{v_d+v_u}

For equal upstream and downstream distances.

Worked Examples

Example 1

Question: Downstream speed is 12 km/h; upstream speed is 8 km/h. Find boat and stream speeds.

Step 1: Average the two speeds to get still-water speed.

u=12+82=10 km/hu=\frac{12+8}{2}=10\text{ km/h}

Step 2: Take half their difference for stream speed.

v=12−82=2 km/hv=\frac{12-8}{2}=2\text{ km/h}

Answer: Boat 10 km/h; stream 2 km/h.

Example 2

Question: Boat speed in still water is 10 km/h and current speed is 2 km/h. Find time for 24 km downstream.

Step 1: Add the current.

vd=10+2=12 km/hv_d=10+2=12\text{ km/h}

Step 2: Divide distance by downstream speed.

t=2412=2 ht=\frac{24}{12}=2\text{ h}

Answer: 2 hours.

Example 3

Question: A boat covers 30 km downstream in 2 hours and 30 km upstream in 3 hours. Find boat and stream speeds.

Step 1: Calculate each journey speed.

vd=302=15,vu=303=10v_d=\frac{30}{2}=15,\qquad v_u=\frac{30}{3}=10

Step 2: Find half the sum and half the difference.

u=15+102=12.5,v=15−102=2.5u=\frac{15+10}{2}=12.5,\qquad v=\frac{15-10}{2}=2.5

Answer: Boat 12.5 km/h; stream 2.5 km/h.

Drill

Try each question before opening its answer.

Question 1

Boat speed 8km/h, stream 2km/h. Find downstream & upstream speed.

  • Show answer

    10, 6

Question 2

Downstream 20km/h, upstream 12km/h. Find boat & stream speed.

  • Show answer

    16, 4

Question 3

A boat covers 36km upstream in 4h. If stream speed is 3km/h, find boat's speed in still water.

  • Show answer

    12 km/h

Question 4

Boat travels 15km downstream and 15km upstream in a total of 4h 30min. Downstream speed 10km/h. Find upstream speed.

  • Show answer

    5 km/h

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