What is the ratio of time for two twins to cross a river at different angles?

[danielatha4]

Homework Statement

Two twins set out to row separately across a swiftly moving river. They have identical canoes, and can row at the same speed in still water. Twin A aims straight across the river but, due to the current, is carried downstream before reaching the opposite bank. Twin B aims her canoe upstream at an angle of 56.0 degrees relative to the riverbank, so as to arrive on the opposite side at a point that is directly across from her starting point.

Calculate the ratio of the time it takes twin A to cross the river to the time it takes twin B to cross.

Determine the direction of twin A's motion, expressed as an angle relative to the downstream direction.

Homework Equations

The Attempt at a Solution

In order to find a ratio of time, I set up update of position formulas.

Twin A:
\DeltaX=V_avg*t

In the case of the x direction I'll call the initial velocity V which should equal Twin B's initial velocity

\DeltaX=V*t

time for twin A:
t=\DeltaX/V

Twin B:
\DeltaX=Vcos34*t
t=\DeltaX/Vcos34

Ratio of A/B for t is cos34, I got this part right.

I don't know how to go about the second part. The drag of the water represents a Force vector, and Twin A start off with a velocity vector in the x direction.

What I do know:
Initial y position=0
Initial y velocity=0

 

[Doc Al]

Part 2 is a vector addition problem, where the two vectors are perpendicular:

The velocity of A with respect to the ground = velocity of A with respect to the water + velocity of the water with respect to the ground​

Hint: Make use of the fact that B travels directly across the river.

 

[danielatha4]

Alright, so can I assume that the vector downstream is Vsin(56)? And the vector to the other side of the stream is V?

Therefore, angle relative to the upstream direction should be arctan(1/sin56)?

Then 180-arctan(1/sin56)= direction relative to downstream direction = 150

But that's wrong. What did I do wrong?

 

[Doc Al]

Alright, so can I assume that the vector downstream is Vsin(56)?

Why sine?

 

[danielatha4]

I assumed sine because it was the y component of the Vector V for twin B and I assumed that since he traveled 0 Y displacement that his Y vector (Vsin56) equaled the vector downstream.

 

[Doc Al]

I assumed sine because it was the y component of the Vector V for twin B and I assumed that since he traveled 0 Y displacement that his Y vector (Vsin56) equaled the vector downstream.

The stream travels along the x-direction, so why use the y-component?

 

[danielatha4]

Sorry, I assumed the river to be in the Y direction. Upstream being higher than downstream.

Doesn't that coincide with the way I solved the first part?

Sorry for the misunderstanding, I'm sure I'm the messed up one though.

 

[danielatha4]

Wait, I see my confusion now. The sin of 56 is the x component. The sin of 34 is the y component. Besides that, did I do the rest of the problem correctly?

 

[Doc Al]

My bad for not realizing that you have the river moving along the y-direction. In any case, the angle is given with respect to the riverbank (the y-axis) and you want the component parallel to the riverbank.

 

[Doc Al]

Wait, I see my confusion now. The sin of 56 is the x component. The sin of 34 is the y component. Besides that, did I do the rest of the problem correctly?

There you go. Other than that, I think it was OK.

 

[danielatha4]

Ok so arctan(1/sin34)=60.8 degrees.

Is this not 60.8 degrees relative to the riverbank from the upstream direction?

So the downstream angle should be 119.2 degrees? but this is wrong...

 

[Doc Al]

Ok so arctan(1/sin34)=60.8 degrees.

Is this not 60.8 degrees relative to the riverbank from the upstream direction?

No, it's relative to the downstream direction. (Rivers flow downstream. )

So the downstream angle should be 119.2 degrees? but this is wrong...

You're just mixing up downstream and upstream. You had the correct answer above.