Speed of a Stream

Question

The speed of a stream is 3 mph.  A boat travels 100 mils upstream in the same time it takes to travel 140 miles downstream.  What is the speed of the boat in still water?

Answer 1

Note:  In this problem we need to understand the difference between actual velocity and relative velocity!

 

 

The actual velocity of the boat is the velocity it would travel in still water or Vb

         

Whereas the  relative velocity of the boat would be the vector sum of the boat and the velocity of flow in the river.

 

Since the boat will travel 100 miles upstream against the flow of the river (Vr = 3mph), its relative velocity would be

 

Vb -Vr or Vb - 3, and the time it took to travel 100 miles upstream would be tu=100/(Vb-3).

 

Likewise, the relative velocity of the boat traveling downstream is Vb + Vr or Vb + 3, and the time it takes the 

 

boat to travel 140 miles downstream is td=140/(Vb=3).

 

Now since the time of travel upstream (tu) equals the time of travel downstream (td) we have

 

tu = td = 100/(Vb-3) = 140/(Vb+3),

 

and solving for Vb we have

 

140 (Vb-3) = 100 (Vb+3)

 

Vb(140-100) = 3(140+100)

 

Vb = 3*240/40 = 720/40 = 18 mph

 

The actual velocity of the boat relative to the shoreline = 18 mph

 

To check this out, we substitute Vb back into the equation 140(Vb-3) = 140(Vb+3)

 

and we have:

 

140(18-3) = 100(18+3)

 

140(15) = 100(21)

 

2100 = 2100 which shows us that our value of Vb is correct!

 

Note: the relative velocity of the boat as it travels upstream is (Vb -3) = 18 - 3 = 15 mph,

 

the relative velocity of the boat as it travels downstream is (Vb + 3) = (18 + 3) = 21 mph, and

 

the time of travel both upstream or downstream is 100/(Vb-3) = 100/(18-3) = 100/15 = 6.66666 hours which we

 

can check by substituting Vb into 140/(Vb+3) = 140/(18+3) = 140/(21) = 6.66666 hours

 

 

Answer 2

Courtney,

 

Here's the trick that lets you solve this type of problem.

s is the boat's speed in still water.

The speed relative to the shore is s-3 going upstream and s+3 going down stream.

 

The time t is the same for 140 miles in one direction and 100 miles in the other direction.

 

d = rt

(1) Downstream: (s+3)t = 140

(2) Upstream:      (s-3)t = 100

 

Solve (2) for s:  (I picked (2) because the solution will have a + sign in it.)

s-3 = 100/t

s = 100/t + 3

 

Substitute this for s in (1):

(s+3)t = 140

[(100/t + 3)+3]t = 140 

(100/t + 6)t = 140

100 + 6t = 140

6t = 40

t = 6.666...

 

s = 100/t + 3              Use this equation again and substitute 6.667 for t

s = 100/6.666 + 3

s = 18 mph

 

Check

(18+3) * 6.667 = 140  miles downstream

(18-3) * 6.667 = 100 miles upstream