Exercise from a foreigner (boat, river and physics)

[Galactico]

Homework Statement

Hello! I'm the foreigner so I'm sorry for my English if it won't be good. I have a big problem with one exercise. I think that I have to use Galilean Transformation but I don't know how to start. There is the content:
A motorboat is flowing by the river upstream with the constant speed. Suddenly lifebuoy has fallen out of the boat. People in the boat discovered it after the time t0 from the moment of its falling out. Then the boat turned back and after the time t (from the turning back) it reached to the lifebuoy which was flowing with the stream. Justify by 2 ways that t=t0: solve this problem in the reference frame connected with the water and in the reference frame connected with the bank of the river. Make all the necessary calculations.
Thanks a lot in advance!

Homework Equations

?

The Attempt at a Solution

I tried to write some equations but with no success :(

 

[AvroArrow]

For the reference frame connected with the water: Let x1 be the position of the lifebuoy at the moment when it falls out of the boat, and x2 be its position when the boat reaches it. Let v be the speed of the boat and u the speed of the stream. x2 - x1 = (v - u)t0x2 - x1 = vtSo, t = t0. For the reference frame connected with the bank of the river:Let x'1 be the position of the lifebuoy at the moment when it falls out of the boat, and x'2 be its position when the boat reaches it. x'2 - x'1 = vt0x'2 - x'1 = vtSo, t = t0.

 

[m2003]

Dear foreigner,

Thank you for reaching out for help with your exercise. I can understand how challenging it can be to tackle a problem in a language that is not your native one. However, I am confident that we can work through this together.

Firstly, let me explain the concept of Galilean Transformation. It is a mathematical tool used to relate the motion of an object in one reference frame to another reference frame that is moving at a constant velocity with respect to the first one. In this case, we can use it to relate the motion of the boat and the lifebuoy in the reference frame connected to the water to the reference frame connected to the bank of the river.

To solve this problem, we can use two different approaches:

1. Reference frame connected to the water:
In this frame, we can consider the boat to be stationary and the river to be flowing downstream with a constant velocity. The lifebuoy will also be moving downstream at the same velocity as the river. When the boat turns back, it will start moving upstream with a velocity equal to the velocity of the river. Therefore, the time taken for the boat to reach the lifebuoy will be the same as the time taken for the lifebuoy to travel downstream from the boat's initial position. This time is t0, as given in the problem statement. Hence, t=t0.

2. Reference frame connected to the bank of the river:
In this frame, the boat will be moving upstream with a constant velocity and the river will be flowing downstream with the same velocity. The lifebuoy will also be moving downstream with the velocity of the river. When the boat turns back, it will start moving downstream with a velocity equal to the velocity of the river. Therefore, the time taken for the boat to reach the lifebuoy will be the same as the time taken for the lifebuoy to travel upstream from the boat's initial position. This time is t0, as given in the problem statement. Hence, t=t0.

To calculate the necessary velocities and distances, we can use the equations of motion, such as v=ut+at^2/2 and s=ut+at^2/2, where v is the final velocity, u is the initial velocity, a is the acceleration, t is the time taken, and s is the distance traveled.

I hope this helps you understand the problem better and gives you a starting point to solve it.