Discover the Flow Velocity of a River with this Boat Trip Calculus Problem

[pringless]

A river flows with a uniform velocity v. A person in a motorboat travels 1.13 km upstream, at which time a log is seen floating by. The person continues to travel upstream for 68.4 min at the same speed and then returns downstream to the starting point, where the same log is seen again. Find the flow velocity of the river. Assume the speed of the boat with respect to the water is constant throughout the entire trip. (Hint: The time of travel of the boat after it meets the log equals the time of travel of the log.) Answer in units of m/s

 

[StephenPrivitera]

First set up a coordinate system. Let the direction of the stream be positive. Set up an equation that will yield the unknown quanitity, in this case, the velocity of the stream (v_s, I assume wrt the shore). Work with the fact that v_s=d_log/dt, where d_log is the displacement of the log downstream (which you know) and dt is the time interval it takes the log to travel this distance. Keep in mind that d_log is positive. Other displacements in this problem will be negative. Now, if you can find dt, you're done (it's actually rather tricky). You will have to consider the boater's velocity in determining dt (pay attention to the direction of v_b!). Give that a shot, and let me know what you come up with.

 

[M98Ranger]

To solve this problem, we can use the formula for velocity: v = d/t, where d is the distance traveled and t is the time taken. In this case, we have two distances (1.13 km upstream and 1.13 km downstream) and two times (68.4 min upstream and 68.4 min downstream). We can set up the following equations:

1.13 km = v * (68.4 min)
1.13 km = v * (68.4 min)

Solving for v in both equations, we get:

v = 1.13 km/68.4 min = 0.0165 km/min

Since we want the velocity in m/s, we need to convert km/min to m/s by multiplying by 1000 m/km and dividing by 60 min/h:

v = (0.0165 km/min * 1000 m/km)/60 min/h = 0.275 m/s

Therefore, the flow velocity of the river is 0.275 m/s. This means that for every second, the river moves 0.275 meters downstream. This boat trip calculus problem gives us a practical application of the velocity formula and shows how it can be used to determine the flow velocity of a river.