A motor boat has speed 30 km/hr in still water. It goes 60 km down stream and comes back in 9/2 hours. What is the speed of the stream?
10 km/hr
Boat and stream problems involve the concept of relative speed. When a boat travels in water, its speed is affected by the speed of the water flow (the stream). There are two main scenarios:
We are given the following details about the boat's journey:
We need to find the speed of the stream (\(V_s\)).
Let the speed of the stream be \(V_s\) km/hr.
The speed of the boat when going downstream is the sum of its speed in still water and the speed of the stream:
Speed downstream (\(V_d\)) = \(V_b + V_s = (30 + V_s)\) km/hr
The speed of the boat when going upstream is the difference between its speed in still water and the speed of the stream:
Speed upstream (\(V_u\)) = \(V_b - V_s = (30 - V_s)\) km/hr
The time taken to travel a certain distance is given by the formula: Time = Distance / Speed.
Time taken to go downstream (\(T_d\)) = \(\frac{\text{Distance}}{\text{Speed downstream}} = \frac{60}{30 + V_s}\) hours
Time taken to come back upstream (\(T_u\)) = \(\frac{\text{Distance}}{\text{Speed upstream}} = \frac{60}{30 - V_s}\) hours
The total time for the round trip is the sum of the time taken for the downstream and upstream journeys:
\(T_{total} = T_d + T_u\)
We are given that the total time is 9/2 hours.
So, the equation is:
\(\frac{9}{2} = \frac{60}{30 + V_s} + \frac{60}{30 - V_s}\)
Now, we solve the equation for \(V_s\):
\(\frac{9}{2} = 60 \left( \frac{1}{30 + V_s} + \frac{1}{30 - V_s} \right)\)
\(\frac{9}{2} = 60 \left( \frac{(30 - V_s) + (30 + V_s)}{(30 + V_s)(30 - V_s)} \right)\)
\(\frac{9}{2} = 60 \left( \frac{60}{30^2 - V_s^2} \right)\)
\(\frac{9}{2} = \frac{3600}{900 - V_s^2}\)
Now, cross-multiply:
\(9 (900 - V_s^2) = 2 \times 3600\)
\(8100 - 9V_s^2 = 7200\)
Rearrange the terms to solve for \(V_s^2\):
\(8100 - 7200 = 9V_s^2\)
\(900 = 9V_s^2\)
\(V_s^2 = \frac{900}{9}\)
\(V_s^2 = 100\)
Taking the square root of both sides:
\(V_s = \sqrt{100}\)
\(V_s = 10\) (Since speed must be a positive value)
So, the speed of the stream is 10 km/hr.
Let's check if a stream speed of 10 km/hr gives the correct total time.
If \(V_s = 10\) km/hr:
4.5 hours is equal to 9/2 hours, which matches the given total time. The solution is correct.
The speed of the stream is 10 km/hr.
| Concept | Formula | Value (with \(V_s = 10\)) |
|---|---|---|
| Speed in Still Water (\(V_b\)) | Given | 30 km/hr |
| Speed of Stream (\(V_s\)) | To find (x) | 10 km/hr |
| Speed Downstream (\(V_d\)) | \(V_b + V_s\) | \(30 + 10 = 40\) km/hr |
| Speed Upstream (\(V_u\)) | \(V_b - V_s\) | \(30 - 10 = 20\) km/hr |
| Time Downstream (\(T_d\)) | Distance / \(V_d\) | \(60 / 40 = 1.5\) hours |
| Time Upstream (\(T_u\)) | Distance / \(V_u\) | \(60 / 20 = 3\) hours |
| Total Time (\(T_{total}\)) | \(T_d + T_u\) | \(1.5 + 3 = 4.5\) hours or 9/2 hours |
| Concept | Formula | Notes |
|---|---|---|
| Speed Downstream (\(V_d\)) | \(V_b + V_s\) | Boat speed + Stream speed |
| Speed Upstream (\(V_u\)) | \(V_b - V_s\) | Boat speed - Stream speed |
| Speed in Still Water (\(V_b\)) | \(\frac{V_d + V_u}{2}\) | Average of downstream and upstream speeds |
| Speed of Stream (\(V_s\)) | \(\frac{V_d - V_u}{2}\) | Half the difference between downstream and upstream speeds |
The concept of relative speed is key in boat and stream problems. When moving with the stream, the stream assists the boat, increasing its effective speed relative to the ground. When moving against the stream, the stream opposes the boat, decreasing its effective speed relative to the ground.
It is important to note that the speed of the boat in still water (\(V_b\)) is its intrinsic speed without any influence from the water flow. The speed of the stream (\(V_s\)) is the speed of the water itself.
If \(V_s > V_b\), the boat cannot move upstream against the current; it will be carried downstream.
A man rows down a river 18 km in 4 hours with the stream and returns in 10 hours. Consider the following statements:
(1) The speed of the man against the stream is 1.8 km/h
(2) The speed of the man in still water is 3.15 km/h
(3) The speed of the stream is 1.35 km/h
Which of the above statements are correct?
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A person can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. What is the speed of the current?
The speed of a boat in still water is 15 km/hr. If it can travel 42 km downstream and 28 km upstream in the same time, then what is the speed of the stream ?
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A boat sails 15 km of a river towards upstream in 5 hours. How long (in hours) will it take to cover the same distance downstream, if the speed of river is one-fourth the speed of the boat in still water?
The speed of boat upstream is 5 kmph. Its speed in still water is 10 kmph. How many minutes will it take to row 25 km downstream?
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The downstream speed of a boat is 20 km/hr and the speed of stream is 4 km/hr. What will be the total time taken by the boat to cover 160 km downstream and 96 km upstream?
The ratio of the speeds of a boat in still water and the speed of the river is 3 ∶ 1. The boat takes 45 minutes for a round trip journey. Find the distance of the whole trip if the speed of the stream is 4 km/hr.