A man can row a distance of 8 km downstream in a certain time and can row 6 km upstream in the same time. If he rows 24 km upstream and the same distance downstream in \(1\frac{3}{4}\) hours, then the speed (in km/h) of the current is:
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Solving the Boat and Stream Problem
This problem involves understanding the concepts of relative speed in the context of boat and stream motion. When a man rows downstream, the speed of the current adds to his speed in still water. When he rows upstream, the speed of the current opposes his speed in still water.
Defining Variables
- Let the speed of the man in still water be \(v_m\) km/h.
- Let the speed of the current be \(v_c\) km/h.
Understanding Downstream and Upstream Speeds
- Speed downstream = Speed of man in still water + Speed of current \( = v_m + v_c\).
- Speed upstream = Speed of man in still water - Speed of current \( = v_m - v_c\).
Note that for upstream motion to be possible, the speed of the man in still water must be greater than the speed of the current (\(v_m > v_c\)).
Setting Up Equations from the Given Information
We are given two pieces of information that allow us to set up equations:
- A man rows 8 km downstream and 6 km upstream in the same amount of time.
- He rows 24 km upstream and the same distance (24 km) downstream in a total of \(1\frac{3}{4}\) hours.
From the first condition (Equal Time):
Time = Distance / Speed.
Time taken to row 8 km downstream is \(T_1 = \frac{8}{v_m + v_c}\).
Time taken to row 6 km upstream is \(T_2 = \frac{6}{v_m - v_c}\).
Since \(T_1 = T_2\), we have:
\(\frac{8}{v_m + v_c} = \frac{6}{v_m - v_c}\)
Cross-multiplying gives:
\(8(v_m - v_c) = 6(v_m + v_c)\)
\(8v_m - 8v_c = 6v_m + 6v_c\)
Rearranging the terms to one side:
\(8v_m - 6v_m = 6v_c + 8v_c\)
\(2v_m = 14v_c\)
Dividing by 2, we get a relationship between \(v_m\) and \(v_c\):
\(v_m = 7v_c\) --- (Equation 1)
This means the man's speed in still water is 7 times the speed of the current.
From the second condition (Total Time):
The total time taken to row 24 km upstream and 24 km downstream is \(1\frac{3}{4}\) hours, which is \(\frac{7}{4}\) hours.
Time taken to row 24 km upstream = \(\frac{24}{v_m - v_c}\).
Time taken to row 24 km downstream = \(\frac{24}{v_m + v_c}\).
The sum of these times is the total time:
\(\frac{24}{v_m - v_c} + \frac{24}{v_m + v_c} = \frac{7}{4}\) --- (Equation 2)
Solving the System of Equations
Now we substitute Equation 1 (\(v_m = 7v_c\)) into Equation 2.
First, find the upstream and downstream speeds in terms of \(v_c\):
- Speed upstream \( = v_m - v_c = 7v_c - v_c = 6v_c\).
- Speed downstream \( = v_m + v_c = 7v_c + v_c = 8v_c\).
Substitute these into Equation 2:
\(\frac{24}{6v_c} + \frac{24}{8v_c} = \frac{7}{4}\)
Simplify the fractions on the left side:
\(\frac{4}{v_c} + \frac{3}{v_c} = \frac{7}{4}\)
Combine the terms on the left side:
\(\frac{4 + 3}{v_c} = \frac{7}{4}\)
\(\frac{7}{v_c} = \frac{7}{4}\)
To find \(v_c\), we can cross-multiply or notice that the numerators are equal, so the denominators must also be equal (assuming \(v_c \neq 0\), which it must be for a current to exist):
\(7 \times 4 = 7 \times v_c\)
\(28 = 7v_c\)
Divide by 7:
\(v_c = \frac{28}{7}\)
\(v_c = 4\)
The speed of the current is 4 km/h.
Verification
If \(v_c = 4\) km/h, then \(v_m = 7 \times 4 = 28\) km/h.
- Upstream speed \( = v_m - v_c = 28 - 4 = 24\) km/h.
- Downstream speed \( = v_m + v_c = 28 + 4 = 32\) km/h.
Check the first condition: Time for 8 km downstream = \(8/32 = 1/4\) hour. Time for 6 km upstream = \(6/24 = 1/4\) hour. This matches.
Check the second condition: Time for 24 km upstream = \(24/24 = 1\) hour. Time for 24 km downstream = \(24/32 = 3/4\) hour. Total time = \(1 + 3/4 = 7/4 = 1\frac{3}{4}\) hours. This also matches.
Final Answer
The speed of the current is 4 km/h.
| Concept | Formula |
|---|---|
| Downstream Speed | \(v_{down} = v_m + v_c\) |
| Upstream Speed | \(v_{up} = v_m - v_c\) |
| Time | Time = Distance / Speed |
Revision Table: Boat and Stream Concepts
| Term | Description |
|---|---|
| Speed in Still Water (\(v_m\)) | The speed of the boat or man in water that is not flowing. |
| Speed of Current (\(v_c\)) | The speed at which the water is flowing. |
| Downstream | Moving in the same direction as the current. |
| Upstream | Moving against the direction of the current. |
| Relative Speed | The speed of an object relative to another object or a medium. In this case, the boat's speed relative to the ground is affected by the water's speed. |
Additional Information: Relative Speed in Fluids
The problems involving boats and streams are practical applications of the concept of relative speed. When an object moves in a fluid (like water or air) that is also moving, its speed relative to a stationary point on the ground (or outside the fluid) is the vector sum of its speed relative to the fluid and the speed of the fluid itself.
- Moving with the fluid (Downstream/Tailwind): The speeds add up. Relative speed = Speed of object in still fluid + Speed of fluid.
- Moving against the fluid (Upstream/Headwind): The fluid's speed is subtracted from the object's speed in still fluid. Relative speed = Speed of object in still fluid - Speed of fluid.
This principle is used not only in boat and stream problems but also in aircraft navigation where wind speed affects the plane's ground speed.
Important Questions from Boat and River
A boat goes 30 km upstream in 3 hours and downstream in 1 hour. How much time (in hours) will this boat take to cover 60 km in still water?
The speed of a motorboat in still water is 20 km/h. It travels 150 km downstream and then returns to the starting point. If the round trip takes a total of 16 hours, what is the speed (in km/h) of the flow of river?
The time taken by a boat to travel 13 km downstream is the same as time taken by it to travel 7 km upstream. If the speed of the stream is 3 km/h, then how much time (in hours) will it take to travel a distance of 44.8 km in still water?
A boat goes 27 km upstream and 33 km downstream in 6 hours. In the same time it can go 36 km upstream and 22 km downstream. How much time will it take to go 36 km upstream and 44 km downstream?
- A boatman can row his boat in still water at a speed of 9 km/h. He can also row 44 km downstream and 35 km upstream in 9 hours. How much time (in hours) will he take to row 33 km downstream and 28 km upstream?
