Calculating Distance and Time with two boats in a river

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Homework Help Overview

The problem involves two boats, A and B, crossing a river with a specified width and current velocity. The boats have different objectives: Boat A aims to minimize distance traveled, while Boat B seeks to minimize time taken. The velocities of the boats and the river are provided, but there is confusion regarding the calculations and the application of vector components in determining the crossing times.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the calculations made for the time taken by each boat to cross the river, noting discrepancies in the original poster's approach. There is an exploration of the need to consider vector components of velocity rather than simply adding or subtracting speeds. Questions arise about how to correctly apply these vector components to the problem.

Discussion Status

Some participants have provided guidance on the importance of vector components in this context, suggesting that the original poster reconsider their calculations. There is an ongoing exploration of the implications of using the correct distances and velocities, as well as the need for clarification on how to apply the concept of vectors to the problem.

Contextual Notes

The original poster has acknowledged a potential error in the distance used in their calculations, indicating a need for clarity on the correct values. There is also a mention of the original poster's confusion regarding the application of vector components, which is central to the problem's resolution.

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Homework Statement



Two boats, A and B, travel with a velocity of 4.90m/s across a river with a width of 72.9m. The river flows with a velocity of 2.50m/s. Boat A travels the shortest distance and boat B travels the shortest time. If both start at the same time, how much time will they take to cross the river?

Homework Equations



There are no equations given, but I was able to use this...
Δd/v(boats)+v(river)=shortest time
Δd/v(boats)-v(river)=shortest distance

The Attempt at a Solution


Boat A: (Shortest Distance)
72.0m/(4.9m/s-2.5m/s)=72/2.4=30s

Boat B:(Shortest Time)
72m/(4.9m/s+2.5m/s)=72/7.4=9.7s≈10.0s

My Answer: Boat A can make it in 30s while Boat B can make it in 10s. I did this on a test and missed all of the points possible. Can anyone please help me find my error ansd reach a resonable answer? Thank You!
 
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tajivie said:

Homework Statement



Two boats, A and B, travel with a velocity of 4.90m/s across a river with a width of 72.9m. The river flows with a velocity of 2.50m/s. Boat A travels the shortest distance and boat B travels the shortest time. If both start at the same time, how much time will they take to cross the river?

Homework Equations



There are no equations given, but I was able to use this...
Δd/v(boats)+v(river)=shortest time
Δd/v(boats)-v(river)=shortest distance

The Attempt at a Solution


Boat A: (Shortest Distance)
72.0m/(4.9m/s-2.5m/s)=72/2.4=30s

Boat B:(Shortest Time)
72m/(4.9m/s+2.5m/s)=72/7.4=9.7s≈10.0s

My Answer: Boat A can make it in 30s while Boat B can make it in 10s. I did this on a test and missed all of the points possible. Can anyone please help me find my error ansd reach a resonable answer? Thank You!

Neither boat travels directly with or against the current (so simply adding or subtracting the current speed to the boat's speed is not correct). You have to consider velocity components (vectors). Also, the distance is given as 72.9m and you've used 72m.
 
Sorry, I meant to type 72.0 in the original problem instead of 72.9.
Can you explain to me what you mean by vector components? I know what they are, I am just confused as to how you can apply them to this problem.
 
tajivie said:
Sorry, I meant to type 72.0 in the original problem instead of 72.9.
Can you explain to me what you mean by vector components? I know what they are, I am just confused as to how you can apply them to this problem.
A boat's velocity with respect to the river can have two components; one directed straight across the river, and one directed up or downriver. The component directed straight across moves the boat in the direction of the far shore. Only the one directed up/downriver can influence the boat's motion up/downriver.
 

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