If you ever swim or paddle upstream, you will notice two things. First, a river’s speed varies a lot. Second, those variations should cause you to pull harder when you hit rapidly flowing water. If you don’t, you will simply make no progress. This puzzle replaces your muscles with a motor, but still asks you to figure out how to trade off energy for time.
Here are the facts:
• You want to go 72 kilometers (km) upriver. • The first 24 km has a downstream speed of 7 kilometers per hour (kmh). • The next 18 km has a downstream speed of 2 kmh. • The last 30 km has a downstream speed of 0 kmh (the river becomes a lake).
You have an electric motor with three settings that can push the boat forward at a water speed of:
• 5 kmh using 1 kilowatt (kW) of power • 10 kmh using 3 kW • 15 kmh using 5 kW
Recall that land speed = water speed - downstream speed. So, for example, if your water speed upstream is 15 kmh but the river has a downstream speed of 2 kmh, then your land speed is 13 kmh.
Warm-up: Suppose you went full speed on all legs of the voyage. How long would the journey take and how much energy would you expend? Solution to Warm-Up Here now are the challenges for you.
- What is the least energy you could use to make the entire trip, assuming you were in absolutely no rush? How would you do it?
Hint: On a lake, you would use the slowest speed, but this may not hold on all parts of the trip.
- Suppose you have a battery that holds 30 kWh. How could you arrange to arrive as quickly as possible without consuming more than 30 kWh? Click here for the solution
Solution to Warm-Up
Here now are the challenges for you.
- What is the least energy you could use to make the entire trip, assuming you were in absolutely no rush? How would you do it?
Hint: On a lake, you would use the slowest speed, but this may not hold on all parts of the trip.
- Suppose you have a battery that holds 30 kWh. How could you arrange to arrive as quickly as possible without consuming more than 30 kWh?
Click here for the solution
