**Team:** 21

**School:** La Cueva High

**Area of Science:** Physics

**Interim:** Definition of the problem:

Secondary schools do not integrate the discussion of drag into standard physics courses and only briefly touch on terminal velocity in AP Physics C; but to what extent does drag forces affect the trajectory of a projectile? Solving the differential force vector equation, we determine that depending on the launch speed of the projectile, the approximation of the range varies drastically. However, all of the previous examples have only dealt with 2D motion, or the effect of only drag and gravity on an object. In the real world, wind is also a contributing factor to the trajectory of the projectile, which pushes projectile motion into 3D. Thus, our project is as follows: given the initial launch velocity, wind velocity, and preliminary physical information about the object and its environment, we will predict the impact point of an object under the influence of the forces of gravity and drag. This can be used in a variety of situations, such as military launch projections, projections for booster rockets in space launches, and crash site predictions for commercial and military enterprises.

Our plan to solve the problem:

1. Build a model that, when given launch velocity and wind velocity, can calculate to a high degree the site that the object will land while taking into account drag force and gravity. However, there is currently no algebraic solution to this differential equation. Thus, we plan to use Euler’s method, used to approximate function solutions to otherwise unsolvable differential equations. However, since Euler’s method is only an approximation, there will inevitably be error in the approximation. To minimize error, we plan to use small step sizes in calculations.

2. Build a model that, when given the location of the landing site, will calculate the launch velocity needed to bring the projectile to that site. To accomplish this, we will implement gradient descent using our part 1 model to refine launch information.

Description of our progress:

A large part of the C++ code for our model has already been written, and tested a number of times. Work still needs to be done to increase efficient with scalability, and more research needs to be conducted to ensure that the numbers and equations that we are working with are as accurate as possible as to ensure exact calculations.

Expected Results:

The most important result that we are trying to achieve is accurate computations of drag force in projectile motion containing a variety of input variables. These computations would contain little of the error present in human computations, and would be accomplished in a fraction of the time. Since our code is based upon preexisting code that is proven to work, we expect for the results to reflect this accuracy. The computation of the results should also have a much quicker runtime than previous code if we are able to correctly scale it down. Then, the result will be code capable of very quickly and accurately calculating the impact point of an object given a variety of different input conditions.

Referenced Citations:

Avriel, M. (2003). Nonlinear programming: Analysis and methods. Mineola, NY: Dover Publications.

David, A. Euler’s Method. Retrieved November 15, 2018, from www.ugrad.math.ubc.ca/coursedoc/math100/notes/mordifeqs/euler.html

Zeleny, E. (2009, April). Wolfram Demonstrations Project. Retrieved November 10, 2018, from demonstrations.wolfram.com/ProjectileWithAirDrag/

Hall, N. (2015, May 5). Forces on a Falling Object. Retrieved December 10, 2018, from www.grc.nasa.gov/www/k-12/airplane/falling.html

Bogley, W. A. Falling Body with Air Resistance. Retrieved December 10, 2018, from oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html

**Team Members:**

David Feng

Luke Xue

Brad Zhang

Blake Watson

**Sponsoring Teacher:** Terri Jones