A river flows with a uniform velocity v. A person in a motorboat travels 1.11km upstream, at which time she passes a log floating by. Always with the same throttle setting, the boater continues to travel upstream for another 58.8min and then returns downstream to her starting point, which she reaches just as the same log does. Calculate the velocity of the river.
To calculate the velocity of the river, we can use the concept of relative velocity. Let's go step by step.
Step 1: Define the variables
Let v be the velocity of the river.
Let v_b be the velocity of the boat (relative to the ground).
Let v_l be the velocity of the log (relative to the ground).
Step 2: Analyze the upstream movement
When the boat is traveling upstream, it is moving against the current of the river. Therefore, the effective velocity of the boat will be the difference between its velocity relative to the ground (v_b) and the velocity of the river (v). So, the effective velocity upstream is v_b - v.
Step 3: Calculate the time taken to travel 1.11 km upstream
We are given that the boat traveled 1.11 km upstream and passed a log floating by. The time taken for this part of the journey can be calculated using the formula:
Time = Distance / Velocity
Since the effective velocity upstream is v_b - v, the time taken to travel 1.11 km upstream is:
t1 = 1.11 km / (v_b - v)
Step 4: Calculate the time taken to return downstream
The boat continues to travel upstream for another 58.8 minutes. In total, the time taken upstream is t1 + 58.8 minutes.
Step 5: Analyze the downstream movement
When the boat is traveling downstream, it is moving with the current of the river. Therefore, the effective velocity of the boat will be the difference between its velocity relative to the ground (v_b) and the velocity of the river (v). So, the effective velocity downstream is v_b + v.
Step 6: Calculate the time taken to return downstream
The time taken to travel downstream from the starting point to where the log is again reached can be calculated using the formula:
Time = Distance / Velocity
Since the effective velocity downstream is v_b + v, the time taken to travel the same distance downstream is:
t2 = 1.11 km / (v_b + v)
Step 7: Set up the equation
We know that the time taken upstream (t1 + 58.8 minutes) is equal to the time taken downstream (t2). Therefore, we can set up the equation:
t1 + 58.8 = t2
Step 8: Substitute the values in the equation
Substituting the values from Step 3 and Step 6 into the equation:
1.11 km / (v_b - v) + 58.8 = 1.11 km / (v_b + v)
Step 9: Rearrange and solve for v (velocity of the river)
To solve for v, we need to manipulate the equation:
1.11 km / (v_b - v) - 1.11 km / (v_b + v) = -58.8
Now we need more information about the velocity of the boat relative to the ground (v_b). If you provide that information, we can help you calculate the velocity of the river.