A fisherman sets out upstream from Metaline Falls on the Pend Oreille River in northwestern Washington State. His small boat, powered by an outboard motor, travels at a constant speed v in still water. The water flows at a lower constant speed vw. He has traveled upstream for 1.65 km when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 12.5 minutes. At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as it is about to go over the falls at his starting point. How fast is the river flowing? Solve this problem in two ways.
(a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v-vw and downstream at v+vw.
(b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems, such as calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets.
Answers
(a) To solve this problem using the Earth as a reference frame, we need to understand the relationship between the boat's speed relative to the water and its speed relative to the Earth.
Let's assume the speed of the boat in still water is v and the speed of the river flow is vw.
When the boat is traveling upstream, its speed relative to the Earth is v - vw, because the speed of the river flow is opposing the motion of the boat.
When the boat is traveling downstream, its speed relative to the Earth is v + vw, because the speed of the river flow is aiding the motion of the boat.
Now let's analyze what happens in the given scenario:
1) The fisherman travels upstream for 1.65 km before realizing his ice chest is missing.
2) He then turns around and starts traveling downstream at the same speed.
3) He catches up with the ice chest just as it is about to go over the falls at his starting point.
Since the boat travels at the same speed relative to the water in both directions, we can say that the time taken to travel a certain distance downstream will be the same as the time taken to travel the same distance upstream.
From the given information, we know that the time taken to travel 1.65 km upstream is 12.5 minutes. Therefore, the time taken to travel the same distance downstream (back to the starting point) is also 12.5 minutes.
Now, let's set up equations to solve for the speed of the river flow (vw).
When the fisherman is traveling upstream:
Distance = Speed x Time
1.65 km = (v - vw) x (12.5 minutes)
When the fisherman is traveling downstream:
Distance = Speed x Time
1.65 km = (v + vw) x (12.5 minutes)
Simplifying these equations, we get:
1.65 km = (v - vw) x (12.5/60) hours
1.65 km = (v + vw) x (12.5/60) hours
Now we have two equations with two unknowns (v and vw). We can solve them simultaneously to find the values.
(b) Alternatively, we can solve this problem using the water as the reference frame.
In this approach, the speed of the boat relative to the water remains constant in both directions. Therefore, we can say that the time taken to travel a certain distance downstream will be the same as the time taken to travel the same distance upstream.
Since the fisherman catches up with the ice chest just as it is about to go over the falls, we can conclude that the distance covered by the ice chest in the time it took to travel 1.65 km upstream is also 1.65 km.
Therefore, by considering the fact that the speed of the boat relative to the water is constant, we can set up an equation:
Distance (covered by the ice chest) = Speed (of the boat relative to the water) x Time (taken to travel 1.65 km upstream)
1.65 km = v x (12.5 minutes)
Simplifying this equation, we get:
1.65 km = v x (12.5/60) hours
By rearranging the equation, we can solve for the speed of the boat relative to the water (v).
Once we have the value of v, we can calculate the speed of the river flow (vw) by substituting its value into either the upstream or downstream equations mentioned in part (a).
Both approaches will result in the same value for the speed of the river flow (vw), which represents the answer to the problem.