Developing a Working Formula for Energy

Hopefully, in exercise 3 you realized that gravity was the force opposing the crow's upward flight. Recall the following definition of force:

where F is force, M is mass, and A is acceleration.

In our case, the acceleration A is the constant acceleration of gravity g = 9.81 m/s2. We will also make the simplifying assumption that the mass of all the crows, and the mass of all the whelks are the same. Thus, the force opposing the crows' flight is the product of two constants; therefore, it is a constant itself, which we will denote k. The distance the crows are travelling, D, is actually the height that they are flying up before they drop the whelks, H. Also, as stated before, work and energy are the equivalent. So instead of W, we use E. Now we can form a working equation for a crow dropping a whelk once, following the steps described above:

Then, if we want to know the energy it takes for a crow to drop a whelk N times, we just multiply to get

We can make one more change to simplify the equation. In this study, we are looking to find if flying to a particular height minimizes the energy a crow uses. The actual value of energy at each height is not important to us. Therefore, we can assume that the constant k = 1, since this will scale each calculation in the simplest possible way. If a particular height gives the minimal energy used, this will still be the case in our new equation: