Downstream

The Stream Boat problems are a frequent part of many important exams. These problems can be split into two main sections. One of them is what we call the downstream problems. In the stream boat problems, a boat goes upstream or downstream. You will have to answer the questions about the speed of the boat and the speed of the river. Here we will see many such examples and try to get as familiar with these concepts as possible. Let us start with the visualization of the downstream problems and try to develop formulae. These formulae will help us establish a method that will accurately and swiftly solve these problems.

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Downstream

In the river-boat problems, we make some assumptions. These assumptions are more like axioms that we have for these problems. A stream or a river can flow in one direction only. It goes either up (opposite to the direction of our boat) or down (in the same direction as the direction in which our boat travels). The speed or the velocity of the stream remains constant at all points. We have to neglect the effects of the eddies or friction or any other factors that may accelerate or decelerate the motion of the boat.

To solve the downstream problems let us first discuss the downstream motion of the boat. The downstream motion means that the boat and the stream or the river are going in the same direction. Therefore the speed of the boat will always be greater than the speed of the river. This is the case when we are rowing the boat to go in the downstream direction. In this case, we have to consider the relative speed or the relative velocity as we will discuss below.

The Downstream Formula

Consider an observer sitting on the banks of a river or a stream. Let us say that this observer measures the speed of the river as u km/hr. A man is rowing downstream and with respect to the river measures his velocity as v km/hr. Then the velocity of the boat as seen from the banks = ( u + v ) km/hr. This is the formula for the downstream boating. This is also known as the relative velocity of the boat with respect to the river bank. Let us see a few examples of this formula.

Let us try to find a formula for the speed of the boat with respect to the river or the speed of the boat in still water. Let ‘u’ be the speed of the boat downstream and ‘v’ be the speed of the boat upstream. Consider the following formula 1/2 [Downstream speed + Upstream speed].

Now the downstream speed of the boat = speed of river + speed of the boat in still water

The upstream speed of the boat = speed of the boat – speed of the river in still water

Using these two formulae in the formula 1/2 [Downstream speed + Upstream speed], we have:

1/2 [(speed of river + speed of the boat) + (speed of the boat – speed of the river)] = 1/2[2 (speed of boat)]

Therefore, 1/2 [u+ v] = speed of the boat in still water. Hence we have found out a formula for the speed of the boat in still water.

Solved Examples For You

Example 1: A man inside a boat rows 9 km in on hour when rowing in still water. In a further part, the stream starts to flow. Now the person takes twice as much time to go upstream as he takes to go downstream. If the distance covered is same, can you find the speed of the stream?

A) 6 km/h               B) 9 km/hr                C) 3 km/hr                      D) 12 km/hr

Answer: The distance is equal, let it be = d. Then let the speed upstream = d/2t and speed downstream = d/t as per the question. Now we have that the speed of the boat upstream: the speed of the boat downstream = 2:1

Let ‘r’ be the speed of the flow of the river and ‘b’ be the speed of the boat in still water. Then the upstream speed of the boat = b – r. Also, the downstream speed of the boat = b + r. According to the given condition:

(b – r)/(b + r) = 2:1

Therefore, upon cross multiplication we have: 2b + 2r = b – r or b = -3r. Since ‘b’ is the speed of the boat in still water, which is = 9 km/hr. Therefore, we have 3r = -(9) or r = -3 km/hr.

Neglecting the negative sign as this is the speed of the river, we have the speed of the stream = C) 3 km/hr.

Example 2: A boat has a speed of 20 km/h downstream and 10 km/h when going upstream. Find the ratio of the speed of the boat in still water to that of the speed of the stream.

A) 1 : 2                  B) 3 : 2                 C) 2 : 3                 D) 3:1

Answer: Here we have to find the ratio b : r. Again the two speeds have been given. As per the question, we can write:

(b + r)/(b – r) = 20/10

2b – 2r = b + r or b = 3r or b:r = 3:1 and the correct optionn is D) 3:1.

Practice Questions

Q 1: If the speed of the stream is 2 km per hour, and the speed of the boat in still waters is 10 km per hour then find the time taken to cover 60 km downstream.

A) 10 hr               B) 5 hr                      C) 20 hr                     D) 30 hr

Ans: B) 5 hr

Q 2: A man goes downstream and covers a distance of 120 km in 5 hours. The speed of the boat is twice the speed of the stream. Then the speed of the stream is:

A) 4 km/hr               B) 6 km/hr                     C) 8 km/hr                     D) 10 km/hr

Ans: C) 8 km/hr