hi I need help with this assignment. The questions are the ones highlighted in the last page. please use excel. and show work thanks

Air Resistance - Numerical Methods using Excel The goal ofthis lab will be two-fold. First, we will explore how air resistance affects the motion of an object by seeing how the position, velocity, and acceleration of an object changes when air resistance is present. Second, we will discuss how numerical methods can be applied to approximate results and see how Excel (or any equivalent spreadsheet program) can be used to determine the approximate values. Air resistance: When we last considered the free-fall motion of an object in the free-fall lab, we found that for our particular experiment the effect of air resistance was minimal. This is not always the case however, so it is necessary for us to have some method of taking it into account. The difficulty with air resistance is that it is extremely difficult to model, so a variety of functional forms are used to approximate air resistance for a given set of conditions. Let's take a look at what sort ofthings affect air resistance. Imagine an object as it falls downward: As object falls, it goes so through a region that is not a vacuum, but instead one that is filled with countless air particles. The force of air resistance is the result of interactions (collisions) with these air particles. If this is the case {and it is), we can see that the force the object experiences should depend on several factors: -The density ofthe medium (air in this case): In the above diagram, the object on the right experiences more air resistance because the air is more dense (i.e. there more particles per unit volume of space) -The size (or more specically the cross-sectional area) of the object: the larger the cross-sectional area, the more air particles it will interact with: In the above diagram, we can see that as the objects move downward, they cover a volume given by the cylinder of cross-sectional area identical to the object. The object on the right (larger cross-sectional area) covers a larger volume of space and will interact with more air particles on the way down. -The \"d rag coefficient\": I I I I O I I I ' . I ' . I I I ' I ' I I C O I I I ' I ' I ' I I ' I ' I I I . I . I I I I I . I 3 C I In the above diagram, although both objects have the same cross-sectional area, the object on the right is more aerodynamic so it experiences less impactful interactions with the air. The collisions with the air particles for the object on the right are more of a glancing type, while the collisions are more \"head-on\" for the object on the left. -Last|y, and most interestingly, it is dependent on the velocity of the object. The faster an object moves, the more air particles it will collide with per unit time, the larger the force of air resistance. The above is where the equation that was previously given for air resistance came from: 1 _ 2 fair resistance _ E pACV Note that all of the factors (density, area, drag coefficient} except velocity are constants for a given system. The velocity is the sole term that changes depending on what the object is doing. In other words, the force of air resistance is a velocitydependent force, Le. a function of velocity: fair resistance = ff\") Here's where it gets interesting: It's easy for us to say that the force is a function of velocity. The problem is determining what the function actually is. The short answer is that we do not know, but the good news is that we can approximate it. This portion is beyond the scope of this course, but one of the topics you'll cover in Calc 2 are Taylor/Maclaurin series. In summary, what it says is that a given function can be expanded into an infinite sum of the form: f"(0) .' 1433(0) 1 _F"'(0) A'+ .\\" +...+ 2! 3! n! f (.\\')=_f (U)+ f' {0).\\' + x" + For our purposes, this means that the force of air resistance can be expanded to take the form: fairremmme = f0?) = a + In: + 112 + (173 + 8174 + ---, where a, b, c, d, e, ..., are constants. The question then becomes what the constants are. The "a" we can argue immediately should be zero. This is because the force of air resistance should be zero when v=0 (i.e. when an object is not moving, there is no air resistance}, so: fairresistance = if\") = by + CV2 + (1193 + 81-74 + The "c\" term we analyzed above and showed was c = ipAC. The remaining terms would typically be experimentally determined for a given object and fluid. For this lab we will ignore how to determine the constants and instead study the velocity dependence by approximating the behavior using numerical methods. The first issue we will address is how many of the "v\" terms we should keep since the Maclaurin series is an infinite sum. We will elect to keep up to the v"2 term and also make the terms negative (since they are opposite the velocity): _ ~ 2 fair resistance _ f0?) \" _b1-7 _ C\" The justification for the above is that it works fairly well to model air resistance. If it did not, we might add the cubic term, and if that did not, we could then add the quartic term, and so on, until it did work. But again, it turns out that keeping just up to the squared term works pretty well. in the above equation we will refer to \"b\" and \"c" as the linear and quadratic coefficients ofair resistance. The last step we will take is to rewrite the v"2 term as v times the absolute value of v: fairresistance = f0\") = b1) _ CVI'UI This is done so that the negative sign is preserved when v is a negative number, otherwise the v"2 would get rid of the negative sign when it was squared. In other words, with the above fix in place, when v>0, the air resistance is negative; and when v0, so air resistance negative i.e. downwards), as it does on the way down (v