The energy required by a fish swimming at speed v to travel a distance

In summary, the energy required for a fish to swim a distance L in a current of speed u is given by the formula E(v) = aL((v^3)/(v-u)), where a is a positive constant. To find the speed of the fish that results in minimal energy expenditure, one must differentiate the function E(v) and find where it is equal to zero. A qualitative sketch of the energy as a function of speed can also be drawn, with a being included as a constant and all values being positive.
  • #1
girlygirl93
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The energy required by a fish swimming at speed v to travel a distance L>0 in a current of speed u>0 is given by

E(v) = aL((v^3)/(v-u)), v>u

where a>0 is a proportionality constant.
a) Find the speed of the fish which results in minimal energy expenditure.
b) Give a qualitative sketch of the energy as a function of the speed of the fish.

I know that I am supposed to isolate v from the equation by using another equation in order to differentiate it, but I don't know how to. As well as a is a constant, and all values are positive. Also when I am drawing the sketch will the value a still be included? How do I include this in a graph?
 
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  • #2


girlygirl93 said:
I know that I am supposed to isolate v from the equation by using another equation in order to differentiate it, but I don't know how to.

Why? If you want to minimize the function E(v) you need to differentiate E with respect to v and find out where it is zero.
 

1. How is the energy required by a fish swimming at speed v to travel a distance calculated?

The energy required by a fish to swim at a certain speed is determined by the fish's body mass, water density, and the drag experienced by the fish as it moves through the water. A common equation used to calculate the energy required is: E = ½ ρ C D A v3 t, where ρ is the density of water, C D is the drag coefficient, A is the frontal surface area of the fish, v is the swimming speed, and t is the time taken to travel the distance.

2. Does the energy required by a fish to swim at speed v vary for different fish species?

Yes, the energy required by a fish to swim at a certain speed can vary depending on the species of fish. This is because different fish have different body shapes, sizes, and swimming techniques, which can affect their drag coefficient and frontal surface area. Additionally, factors such as water temperature and water depth can also impact the energy needed for swimming.

3. How does water temperature affect the energy required for a fish to swim at speed v?

Water temperature can have a significant impact on the energy required for a fish to swim at a certain speed. Warmer water is less dense, which means there is less resistance for the fish to move through. As a result, the fish may require less energy to swim at a certain speed compared to swimming in colder water.

4. Is there a difference in the energy required for a fish to swim at speed v in still water versus moving water?

Yes, there is a difference in the energy required for a fish to swim at a certain speed in still water versus moving water. In still water, the fish must create its own movement, which requires more energy. In moving water, the fish can use the water currents to help propel itself, reducing the amount of energy needed for swimming.

5. Can the energy required for a fish to swim at speed v be reduced?

Yes, the energy required for a fish to swim at a certain speed can be reduced through various methods. This includes reducing the drag coefficient by altering the fish's body shape or swimming technique, using water currents to assist in swimming, or adjusting the fish's swimming speed to the most efficient level. Additionally, maintaining a healthy and well-nourished fish can also help reduce the energy needed for swimming.

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