Calculating Distance and Time with two boats in a river

[tajivie]

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!

 

[gneill]

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.

 

[tajivie]

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.

 

[gneill]

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.

 

[asteriatic]

I would like to clarify that the equations used in this solution are not the correct equations for calculating distance and time in this scenario. The correct equations to use are:

Distance = Velocity x Time

Time = Distance / Velocity

Using these equations, we can calculate the time it takes for each boat to cross the river:

Boat A: Distance = 72.9m, Velocity = 4.9m/s
Time = 72.9m / 4.9m/s = 14.9s

Boat B: Distance = 72.9m, Velocity = 7.4m/s (4.9m/s + 2.5m/s)
Time = 72.9m / 7.4m/s = 9.8s

Therefore, Boat A will take 14.9 seconds to cross the river while Boat B will take 9.8 seconds. The error in the previous solution was using the incorrect equations, resulting in incorrect answers. As a scientist, it is important to always use the correct equations and double check calculations to ensure accuracy.