How Do I Go About Solving These Two Equations Differently

What is the difference between this two equations : y=-2x and y=|-2x|?

y=|-2x| looks like two different equations:Whenx>=0: y=2xx<0: y=-2x|| is a magnitude operator that makes it’s parameter positive. y=-2x is a straight line, starting from the top left, passing (0,0). y=|-2x| will bounce that off the x axis.Computational Knowledge Engine

Why do the two situations have different formulas for the normal force at 5:56 in the video? Can't you just split the components of the normal/Fg differently and get a different formula for the normal?

The gravitational force is downwards, while the normal force is perpendicular to the ramp. In the first case, the “battle” is between the component of gravitational force along the ramp, vs. frictional force. The frictional force is proportional to the normal force and fully parallel to the ramp (and it's unfortunate that the video presenter did not include the coefficient of (static) friction, and better vector notation, so that this point is clearer).Thus it is a component of the gravitational force, mg cos (theta), that must match the full normal force (multiplied by the coefficient of static friction).The second case also doesn’t set up the full picture. However, the basic difference that is emphasized is the “battle” between a directly vertical gravitational force and the vertical component of the normal force. [Although due to the centrifugal force on the curve road and the ramp angle, the final function will be more complicated—so far this is just a factor.]So in this case, it is directly mg that must match just the vertical component of the normal force, (Fny =) Fn cos (theta).Therein lies the difference in the normal force equations, but again, these are not yet the full formulation of the equations need to solve the two problems.

How does an identity differ from an equation? Provide a few examples.

An equation is essentially a statement of the equality of two mathematical expressions . An identity is an equation where the two mathematical expressions are equal for all the values of their variables.Here are some more definitions and explanations :In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables.Each side of an equation is called a member of the equation. Each member will contain one or more terms. The equation,[math]{\displaystyle Ax^{2}+Bx+C=y}[/math]has two members: [math]{\displaystyle Ax^{2}+Bx+C}[/math] and [math]{\displaystyle y}[/math]. The left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, and C.Source : EquationIn mathematics an identity is an equality relation A = B, such that A and B contain some variables and A and B produce the same value as each other regardless of what values (usually numbers) are substituted for the variables. In other words, A = B is an identity if A and Bdefine the same functions. This means that an identity is an equality between functions that are differently defined. For example, [math]\displaystyle (a+b)^2 = a^2 + 2 a b + b^2[/math] and cos2(x) + sin2(x) = 1 are identities. Identities are sometimes indicated by the triple bar symbol ≡ instead of =, the equals sign .Source : Identity (mathematics)See also the following related link :What is the difference between an identity, an equation and a conditional equation?

What happens If two liquids of different densities are mixed?

There appear to be 3 possibilities;1. If the liquids are immiscible, and they are not stirred, or only stirred gently, they will seperate into 2 layers, with the less dense floating on the more dense liquid. There will be a small amount of dissolving at the interface, but this will quickly reach equilibrium, with very small concentrations of each liquid dissolved in the other.2. If the liquids are again immiscible, but they are stirred very vigorously, then you can get an emulsion, where tiny droplets of the smallest phase are held, suspended in the other bulk phase liquid. Examples include milk, mayonnaise and butter. Some are naturally stable, whilst others need an emulsifier to keep the droplets dispersed and in suspension.3. The liquids are miscible, and completely dissolve in each other. Now, if there is a volume, V(1), of liquid 1, with a density of d(1), mixed with liquid 2, of volume, V(2), and density,d(2), it seems that the following formula ought to give D, the density of the mixture;D = {d(1)V(1) + d(2)V(2)} / {V(1) + V(2)}The problem with this formula is that it assumes that the total volume,V(T) = V(1) + V(2), but this might not be the case. V(T) could be smaller or larger than V(1) + V(2), and there’s no way that I know of to calculate what it is. So, the only way to find out is to measure it, experimentally.

How is calculus different from algebra?

Answering your question in a very primitive way: algebra is like an extension to arithmetic. Instead of numbers you are dealing with letters. The main idea is that by adding two to three you performed one addition from the possible infinite numbers. by using a+b you may have used any of all possible numbers. Of course my explanation is a bit of an over simplification, but gives you the basic idea.Turning to calculus: you are dealing with two distinct types. One is called differential calculus and the other is integral calculus. Again, I can’t give you a 5 minute lecture on those, but give you the basic idea. By differential calculus we also use the term, derivative of a function. The derivative of a function is used to find the rate of change of a function. How quickly does it change as you trace the function, so to speak. It can also give you the equation of the tangent to a function. Well, enough of differential calculus. Now let us turn to integrals. Imagine a function drawn as a curve of some kind in a coordinate system. Suppose you want to find the area under that function . The integral calculus will give you that. Of course you have to identify the limits where you want the area to be calculated. Similarly, instead of two dimensional functions you can go to three dimensions, and instead of areas, volumes can be calculated.I guess you can see from this extremely brief introduction the difference between calculus and algebra.

How do I calculate the time for two moving, accelerating objects to reach one another?

The kinematics formulas for constant acceleration arex = x0 + v0 * t + 1/2 a * t^2 and v = vo + a * twrite an x formula for object A, and an x formula for object B. For the objects to meet, they have to be in the same place at the same time, so set the x’s equal to each other, and solve for the time (since it is the same t on both sides.)