Q. 1: Two boats, travelling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide?
(1) 1/12 (2) 1/6 (3) 1/4 (4) 1/3
Now this is a very very easy question. If one pauses for a moment before starting to find the time taken to collide; one pauses long enough to apply pencil to paper, one can realise that we can work-backwards from the time that they collide i.e. play the scenario backwards i.e. if they start together and move apart, how far apart will they be in 1 minute.
And since they are in a stream, assuming the speed of stream as R, we get the answer as:
Dist = Relative Speed × Time =
= ¼ km.This article is about highlighting a simple point but very potent point. But before we give away the mantra, try these three questions, each being a build-up of the earlier one:
Q. 2: Two boats are at the same point in a stream with speed of current being 4 kmph. One boats starts travelling upstream at speed of 10 kmph and at the same time, the other boat starts travelling downstream at speed of 6 kmph. Both of them travel apart for 4 hours and then reverse their directions and are now travelling towards each other. From now onwards, in how much time will the boats meet?
Q. 3: Two boats start travelling from the same point in a stream, one upstream and one downstream. They travel at their individual uniform speeds, which need not be the same. After 4 hours, they reverse their directions and now travel towards each other. From now onwards in how much time will they meet?
Q. 4: Two boats start travelling from the same point in a stream, one upstream and one downstream. They travel at their individual uniform speeds, which need not be the same. After 4 hours, they reverse their directions and now travel towards each other. At the same time the stream also reverses its direction of flow i.e. the boat which was travelling upstream earlier is even now travelling upstream. From now onwards in how much time will they meet?
Hoping that you would have taken time to solve the above (if not spend some thoughts on the above before reading ahead), here is the mantra ……
When all moving objects are in the same reference frame, say in a stream, the presence of stream is irrelevant.
Quite a few students would have applied pencil to paper while solving Q. 1, simply because values were given. And when you must have given thought to Q. 2, you would have realised that the values just do not matter. Since they are travelling apart for 4 hours, they would take 4 hours to meet. That’s it. The stream, their individual speeds, the fact that one is going downstream and other upstream, etc, just do not matter.
Still not convinced?
When two boats travel in opposite direction in a stream, one with speed S1 travelling downstream and other with speed S2 travelling upstream, their relative speed is (S1 + R) + (S2 – R) = S1 + S2 i.e. their relative speed is same as if they were on land.
And if they were both moving upstream, their relative speed would have been (S1 – R) – (S2 – R) = S1 – S2, again same as if they were on land.
And if they were both moving downstream, their relative speed would have been (S1 + R) – (S2 + R) = S1 – S2, again same as if they were on land.
Logical Interpretation
Consider you and your friend walking towards each other or chasing each other but in a train. But the train is such that all windows and walls are covered with black curtain, nothing of the outside world is visible. For the two of you, the universe, the reference frame is just the train. With no observation of outside world, whether the train is moving, with what speed, in which direction or whether all these factors keep changing does not matter at all.
Another way to look at it is consider any other question of Time Speed Distance, police catching thief, car overtaking bus, a race, etc. In all these scenarios, all the objects are travelling on the earth. And isn’t earth also a vast vast ‘stream’ with its own speed? Say if the earth starts rotating faster or changes the direction of rotation, would you even notice it?
Now it should be clear that in all the three questions, the boats are travelling apart for 4 hours at relative speeds of S1 + S2 and when moving towards each other, again their reltive speed will be S1 + S2. And hence, they would take 4 hours to meet. From the above logical explanation you would have realised that even in the third question, even though the stream changes direction, the time taken to meet will still be the same, 4 hours. Even if the stream changes directions every now and then, it hardly matters.
Relevance of the stream
So what matters because of the stream? What matters is the point at which they meet, w.r.t. to the original point of starting. And this is, also, relevant only from a spectators point of view, a spectator who is standing on land and not a part of the stream. Take some time to ponder about the point where they meet. We will take this up in the next article.
And till then, two supposedly difficult questions based on the above ……
Q. 5: A boat is moving in a stream. While going downstream it crosses a wooden piece. After 1 hour, it reaches the port and start traveling in the opposite direction. Now after moving for 1 hour, it crosses that wooden piece again which is 8 km downstream of the place where it was encountered earlier. Find the velocity of the stream?
a) 4 kmph b) 8 kmph c) 16 kmph d) 12 kmph
Q. 6: While rowing a boat upstream in a river, I accidentally drop my hat in the river but do not realize it then. After further 10 minutes of rowing upstream, I realize the loss of the hat and immediately turn back. I catch up with the hat at a distance of 100 meters downstream from the place I had dropped the hat. Find the speed of the current.
a) 5 m/s b) 1/12 m/s c) 10 m/s d) 1/6 m/s
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