Faster and faster and!

They reach the opposite bank at the same time. Compare their speeds.


Statement

There are 3 boats A \color{#D61F06}{A} , B \color{#20A900}{B} and C \color{#3D99F6}{C} on a river . They travel with velocities v A v_A , v B v_B and v C v_C respectively in still water. They are on one bank of the river. They have to reach the other bank. The speed of stream is v. They travel making angles 120°, 90° and 60° with horizontal w.r.t. stream respectively. They start at the same time and at the same distance from the bank. River is of constant width.


All of my problems are original


Difficulty: \dagger \dagger \color{grey}{\dagger} \color{grey}{\dagger} \color{grey}{\dagger}

v B < v A = v C v_B \lt v_A = v_C v A < v B < v C v_A \lt v_B \lt v_C v A = v B = v C v_A = v_B = v_C v A < v B = v C v_A \lt v_B = v_C v A < v C < v B v_A \lt v_C \lt v_B v C < v B < v A v_C \lt v_B \lt v_A v B < v A < v C v_B \lt v_A \lt v_C v A = v B < v C v_A = v_B \lt v_C

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2 solutions

Aryan Sanghi
Jun 18, 2020

Let the width of the river be D

Now, considering their speeds on y-axis i.e. perpendicular to the motion of stream

v A y = v A s i n 60 ° v_{Ay} = v_Asin60°

v B y = v B v_{By} = v_B

v C = v C s i n 60 ° v_{C} = v_Csin60°


Now, as they reach the opposite end at the same time

d v A y = d v B y = d v C y \frac{d}{v_{Ay}} = \frac{d}{v_{By}} = \frac{d}{v_{Cy}}

1 v A y = 1 v B y = 1 v C y \frac{1}{v_{Ay}} = \frac{1}{v_{By}} = \frac{1}{v_{Cy}}

v A y = v B y = v C y v_{Ay} = v_{By} = v_{Cy}

v A s i n 60 ° = v B = v C s i n 60 ° v_{A}sin60° = v_{B} = v_{C}sin60°


Now, as s i n 60 ° < 1 sin60° \lt 1 , so the order is v B < v A = v C \color{#3D99F6}{\boxed{v_{B} \lt v_A = v_C}}

I don't know trigonometrey, but I solved using common sense!

Vinayak Srivastava - 11 months, 4 weeks ago

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That's what was the real solution. But I can't write it, you know in the solution.

Aryan Sanghi - 11 months, 4 weeks ago

@Aryan Sanghi - You also have to mention that all the boats starts at the same time and from the same distance from the bank, the question will then be perfect.

Zakir Husain - 11 months, 4 weeks ago

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I'll mention, but isn't that understood?

Aryan Sanghi - 11 months, 4 weeks ago

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It isn't understood. Logically it is not always necessary.

Zakir Husain - 11 months, 4 weeks ago

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@Zakir Husain Yes you're right. Thanku.

Aryan Sanghi - 11 months, 4 weeks ago

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@Aryan Sanghi Also, adding that the river is of constant width could be nice. It's implicit because the problem can't be solved without that assumption but clarity cannot hurt..

alazrabed . - 11 months, 3 weeks ago

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@Alazrabed . Thanku. I have updated the question.

Aryan Sanghi - 11 months, 3 weeks ago

uhhhhhhhhhhhh

Dave Owens - 11 months, 4 weeks ago

What do it mean but "they travel with velocities in still water" but "the speed of stream is v"? Is the water flowing or completely still?

Boon Yin Lee - 11 months, 4 weeks ago

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Means in still water, they have those velocities or in other words the velocity is w.r.t. river.

Aryan Sanghi - 11 months, 4 weeks ago

i dont like the drrawings

Ming Rong - 11 months, 3 weeks ago

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Sorry, I'll try to be better next time.

Aryan Sanghi - 11 months, 3 weeks ago

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no i was joking i didnt even expect u to see it

Ming Rong - 11 months, 2 weeks ago

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@Ming Rong Ohk, no problem. :)

Aryan Sanghi - 11 months, 2 weeks ago

btw this is ming rongs son

Ming Rong - 11 months, 2 weeks ago

I don't think this is the correct explanation.

Ankit Udania - 11 months, 1 week ago

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Actually this is a pretty correct explanation, and I am confident about it. Feel free to share your explanation if you found mine incorrect. :)

Aryan Sanghi - 11 months, 1 week ago

The only thing that matters for the boats to reach the other side of the river is their velocity(direct speed) towards the other side. Boat A goes in a straight line(90 degrees) so it has the shortest distance to the other side. Boats B & C go in the same tilted angle ,the only difference is that one is tilted to the right and the other is tilted to the left( |90-X| ), so they require to go through a longer distance than boat A. All boats reach the end at the same time which means that boats B & C are faster than boat A (distance/time=speed) and since boats B & C went through the same distance they have the same speed. In short : velocity of boat A < velocity of boat B = velocity of boat C

you should mention that the velocity you’re talking about is wrt the river

Lowell Chen - 11 months, 1 week ago

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