A Motorboat Traveled 35 km Upstream on a River and Upstream for 18 km
A motorboat traveled 35 km upstream on a river and then up an adjacent stream for 18 km, spending 8 hours on the entire trip. On the return trip, it took only 30 minutes. During both parts of the trip, the speed of the current was 1 km/hour slower than the upstream speed.
What was the speed of the boat in still water?
The answer is: While the speedometer on the boat may read 6 m/s, its actual speed with respect to an observer on the shore will be greater. This is because when a boat is in still water, the motion of the boat relative to the boat’s speedometer (the boat’s velocity with respect to the water) is proportional to its speed with respect to the speedometer.
Part a of this problem asks “How long does it take for the boat to travel across a 120-m wide river?” It requires 20 s for the boat to cross the river. It also drifts downstream during this time.
This is because the current carries the boat downstream for the Distance B; and that component of the motion is what affects how long it takes to travel the distance directly across the river. However, in order to calculate the length that the boat traverses downstream, we must use a speed value that represents the average speed that the boat is traveling in its direction perpendicular to the boat’s motion across the river.
What should be the value that we choose to represent this average speed? Should we choose a value of 3 m/s, 4 m/s, or 5 m/s?
We can find this average speed by substituting the boat’s current velocity in the numerator of the equation. We can also substitute the resultant velocity in the denominator of the equation; but in either case, it must be the same average speed that we used to calculate the time to travel the distance directly across the river, and we must use that same value in the equation for the distance downstream as well.
Then, we can solve for the upstream and downstream speeds by plugging in each of these values into the equation. This is what we do in the example above.
Part b of this problem asks “How far upstream does the boat go when it leaves at 8:00 am and returns at noon?” It takes a total of 4 hours for the boat to travel upstream. It also takes a total of 4 hours for the motorboat to travel upstream when it leaves at 8:00 am and returns to its starting point.
This can be done in the following way: We can write out two equations: one for the upstream speed and the other for the downstream speed; or we can plug each of these equations into the diagram below. Once we have solved the upstream and downstream rates, we can then multiply them together to find the distances in kilometers.
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