Skydiver Quadratic Equations

In the second problem, we are also given the values of gravity, g; the mass of the object, m; the air density, r; the surface area of the object, A; and the coefficient of air resistance, C. Also we know the time at which the parachute is opened. We are also given the initial values of the velocity, in meters per second, and the height, h.

The value r is in kg/m3. The area is a function of time due to the fact that the parachute opens at a certain time, which is 7 seconds, and the surface area will change due to the fact that a parachute will open. The value of mass is in kg and gravity is in m/sec2. The surface area is expressed in m2. Unlike the previous problem where we had to find the graphs of velocity and height vs. time, and answer several questions using those graphs, there is now another factor we must include, a camera. The camera is located on the ground 100 meters from a point directly beneath the path of the skydiver and it is following the falling objects'(the skydivers') descent and we must find the rate at which the camera angle between the height of the object and 100 meters from a point directly beneath the path of the skydiver changes. The camera angle is represented by the variable x. The problem asks us to find the rate of change of the angle at time t=4 and t=10 seconds.

The function for the angle measure in radians is

In order to find the rate of change of the angle measure, the tan inverse of both sides must be taken and then the entire function must be differentiated which will result in:

Where t is time, x is the angle measure, and v is velocity.

Before we can determine the rate of change of the angle, we must first determine what the angle measure actually is at the desired time, t. To find this, we must simply modify the function for the angle measure in radians. We must also use the height function and velocity function in the first problem so we can properly use Euler's method.

Where i is the number of iterates of deltat, ti is the function which will give us all of the t values we need. k(t) is the function that brings together all the values we were given earlier in the problem to determine the upward air resistance. Derv(t,v) is the derivative of velocity and is crucial in order to determine the velocity using Euler's Method. Also, the height function hi is also a function which, if used with Euler's Method, will give us the height at our desired time, if we use Euler's method.

To find the rate of change of angle at t=4 seconds we must plug our desired values into the above function. But before we can plug our values