D for air resistance lab.pdf X DOE Air resistance and Euler's Methoc X + X C File | C:/Users/dolap/Downloads/Air%20resistance%20and%20Euler's%20Method(1).pdf . . . 3 of 8 Q + Page view A Read aloud T Add text Draw Highlight Erase | | P Equation 6 is a beast-what is called a second order nonlinear differential equation. Unlike an algebraic equation whose solution is a number, a differential equation has a solution which is a function. There is some function x(t) that satisfies (6), which describes the sky diver's position over time exactly. It turns out that this equation can be solved exactly by some clever O mathematical guesswork. At some point sooner or later in your undergraduate studies you will take a course in differential equations and learn some of the tricks involved in solving these sorts -+ of equations. In many cases of physical interest, though, the force law is more complicated, and we soon run out of clever tricks. Mathematical analysis is not quite the grand thing it is said to be; it solves only the simplest possible equations. As soon as the equations get just a shade more complicated, they cannot be solved exactly. (For example, if the resistive force happened to be proportional to the cube of the speed, the equation can be proved to be unsolvable.) Numerical Solution (Euler's Method): This is where numerical techniques often come jump in to save the day-and even though it is possible to solve eqn 6 without resorting to this technique, since you don't have the mathematical background to do it, we might as well consider that option off the table. The numerical method described below can take care of any equation of physical interest and is nicely adapted to computers. The tradeoff (there is always a tradeoff) is that the numerical method is not exact; it is only an approximation to the true motion. Suppose that at a given time / the object is at position x and has velocity v. What will be the subsequent motion? If we can figure out what the position and velocity are at a slightly later time, 1 + At, then the problem is solved, for then we can start with the initial condition and compute how it changes for the first instant, the next instant, the next, and so on. In this way we gradually chase the object through its motion. Let's take a familiar constant acceleration equation (x = x; + vit + 1/2at?) and look at how to get a new position x at a very small time interval At later, ED if we know info about the object's velocity and location at time t, playing the role of v; and X;: x(1 + At) = x(1) + v(t) . At + - a(At) (7) 2 68.F 8:16 PM Mostly sunny 7/13/2022 1D for air resistance lab.pdf X DE Air resistance and Euler's Methoc X + X C File | C:/Users/dolap/Downloads/Air%20resistance%20and%20Euler's%20Method(1).pdf . . . 3 of 8 Q + Page view A Read aloud T Add text Draw Highlight Erase | | P (A - 1 1 Vil 1 72at ) anu look at now to get a new position A at a very sman ut merval z Tate, if we know info about the object's velocity and location at time t, playing the role of v; and x;: x(1 + At) = x(1) + v(1) . At + - a(At)2 (7 ) In this particular situation, the velocity v does not have a chance to change significantly during the very small time interval At. Therefore ~ a(At) is insignificant compared to the other terms in eqn 7: x(1 + At) = x(t) + v(t) . At (8) So we are saying the velocity is approximately constant over the short time interval At (from t to t + 41). This is obviously not true unless the force is zero during this time, but we hope if At is small enough, the velocity does not change too much so the equation is still usefully accurate.' + You will employ this trick often in advanced courses-- "keeping the change to first order". We keep only the first and dominant modification to x(t); if At is a small number, then (At) is even smaller; we throw second order At term away. 70 Now what about the velocity? We can't keep it constant forever! There really is an acceleration. In order to get the velocity later, the velocity at time t + 4, we need to know how OneDrive ... X the velocity changes, that is, we need to know the acceleration. By an argument analogous to the one for position we can approximate the velocity at the later time t + 4, by subbing into another Screenshot saved familiar constant acceleration eqn (v = vi + at): The screenshot was added to your OneDrive. V(t + At) = v(t) + a(t) . At (9) 68.F 8:16 PM Mostly sunny O 7/13/2022 1E I @ airreslstancelabpdf XI E Auresustanceand Euler'sMethc > m ) the force of air resistance approaches the E skydiver's weight and the net force on the skydiver approaches zero. + (a) Is your graph of acceleration vs. time consistent with this? Explain. Then hand-sketch\" onto the same graph the \"no air resistance" case. (b) Is your graph of velocity vs. time consistent with this? Explain. Then handasketch' onto the same graph the \"no air resistance\" case. (C) Is your graph of position vs. time consistent with this? Explain. Then hand-sketch" onto the same graph the \"no air resistance\" case. *Yuu can do this part in Excel instead of hand sketch. 1: f One Drive ' r graphs? Then explain 4. How would changing the values of x(l)) and v(D) in table 111 affect you . _ n a :10 E. S'lSPM 1 0312022 E I @ airreslstancelabpdf XI E Auresustanceand Euler'sMethc >