River crossing and relative velocities

• Egoyan
In summary, a boat crossing a river with a speed of 12km/h relative to water and a river with a uniform speed of 6 km/h due east relative to earth. The boat's speed relative to a stationary observer is 10.39 km/h due north, and the boat should be heading at a direction of 120° from the positive x-axis to reach an opposite point directly across the river. The boat's velocity can be calculated using the formula Vbe = Vbw + Vwe, where Vbe is the boat's velocity relative to the earth and observer, Vbw is the boat's velocity relative to the water, and Vwe is the water's velocity relative to the earth. The magnitude of Vbe can
Egoyan
Hi everyone,

I've been having difficulties with this problem for a while. Here is my best attempt at solving it. If there's anything wrong, I honestly can't figure it out :). I would appreciate if anyone could go over it quickly and tell me if/what I did wrong.

Homework Statement

A boat crosses a wide river with a speed of 12km/h relative to water. The river has a uniform speed of 6 km/h due east relative to earth.
(a) Determine the speed of the boat relative to a stationary observer.
(b) In what direction should the boat be heading to reach an opposite point directly across the river?

Homework Equations

I've set up the velocities as such:
Vbe = Velocity of the boat relative to the Earth (and observer),
Vbw = Velocity of the boat relative to the water, and
Vwe = Velocity of the water relative to the earth.

Therefore, Vbe = Vbw + Vwe

The Attempt at a Solution

After drawing a picture of the situation, I've determined that

Vbw = (-12sinΘ i + 12cosΘ j) km/h
Vwe = (6i + 0j) km/h
Vbe = (0i + Vbe j) km/h ← This is something I'm not certain of. Am I right to assure that since we want to go directly across the river, relative to the earth, this vector should have a 0 i-component?

Using Vbe = Vbw + Vwe,

1) -12sinΘ + 6 = 0
→ Θ = 30° (counterclockwise from positive y-axis)

2) magnitude of Vbe = 12cosΘ = 12cos30 = 10.39

To answer a) using the above, Vbe = (0i + 10.39j) km/h, or 10.39 km/h due north relative to the observer.
b) Direction should be 120° from positive x-axis.

Have I made any mistakes somewhere? For some reason, I had quite a hard time visualizing this problem.

Egoyan

Your answer and reasoning look right for b). Part a) is unclear - you cannot answer it without assuming a relative angle.

Hi,

Thanks for taking the time to look at the problem.

That's what I thought too - and that's part of why I had such a hard time visualizing it, I believe. But this was an exam question (I am 100% certain of the phrasing, I've got a copy of it with me), and I was wondering what was the deal here...

Some of my classmates were illustrating the problem as a triangle, setting it up such that the hypotenuse would be the resultant velocity relative to the observer; that is v=√(12^2+6^2) = 13.4 km/h. Anyone knows if this could be right?

Thanks again,

Egoyan

Egoyan said:
Hi,

Thanks for taking the time to look at the problem.

That's what I thought too - and that's part of why I had such a hard time visualizing it, I believe. But this was an exam question (I am 100% certain of the phrasing, I've got a copy of it with me), and I was wondering what was the deal here...

Some of my classmates were illustrating the problem as a triangle, setting it up such that the hypotenuse would be the resultant velocity relative to the observer; that is v=√(12^2+6^2) = 13.4 km/h. Anyone knows if this could be right?

Thanks again,

Egoyan
Yes, I suspect part a should have said "if the boat heads straight across relative to the water". This would make 13.4 correct.

Ah, yes, I see. Thanks a lot!

1. How do you calculate the relative velocity of an object crossing a river?

The relative velocity of an object crossing a river can be calculated by adding the velocity of the object in still water to the velocity of the river current in the same direction, or subtracting the two velocities if they are in opposite directions.

2. What is the best strategy for crossing a river and accounting for relative velocities?

The best strategy for crossing a river and accounting for relative velocities is to aim upstream and at an angle towards the opposite bank. This will help to counteract the downstream drift caused by the river current.

3. How does the width of a river affect the relative velocity of an object crossing it?

The width of a river does not directly affect the relative velocity of an object crossing it, as long as the object maintains a constant velocity. However, a wider river may have a stronger current and therefore a greater impact on the object's velocity.

4. Can the relative velocity of an object crossing a river change during the crossing?

Yes, the relative velocity of an object crossing a river can change during the crossing if the object's velocity or the river current changes. This can also be affected by the shape of the river and any obstacles in the way.

5. How does the angle at which an object crosses a river affect its relative velocity?

The angle at which an object crosses a river can affect its relative velocity by changing the direction of the river current's impact on the object. For example, crossing at a right angle to the current will result in a greater impact and therefore a greater change in velocity compared to crossing at an angle against or with the current.

• Introductory Physics Homework Help
Replies
21
Views
342
• Introductory Physics Homework Help
Replies
8
Views
1K
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
2K
• Introductory Physics Homework Help
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
11
Views
2K
• Introductory Physics Homework Help
Replies
30
Views
3K
• Introductory Physics Homework Help
Replies
6
Views
3K
• Introductory Physics Homework Help
Replies
6
Views
998