Averages in Projectile Motion with Linear and Quadratic Drag

In this work, we calculate the averages of the flight time, range, maximum height, area under the trajectory, length of the trajectory, average speed and average velocity over the launch angle in projectile motion with air drag. We consider two common air drag models: linear and quadratic. For the linear drag model, we incorporate existing analytical solutions of the motion into the integrals to compute the averages. The average maximum height yields an expression involving a combination of dilogarithm functions, which is also expressed in terms of the Legendre chi function of order two or, alternatively, Ramanujan's inverse tangent integral. We derive analytical expressions that describe how the averages behave in the weak and strong drag limits. For the quadratic drag model, we compute the averages based on numerical solutions of the equations of motion. We present the results as functions of the drag constant in each model.