2021-12-19

Kingbird Furniture company

49

Here is version using tabular or array (saves you having to go into and out of math mode for each cell) :

**1. Large tilde over math symbol with automatic horizontal positioning**

I can suggest to use the lite version of mtpro2 that you can find to CTAN. Here for example there is a recent documentation (see pag. 14). Is you will use this MWEyou will obtain this output. The best tilde is with the command wwtilde a good solution to center it with any letter. Personally I suggest to leave the same size (or dimension) of "tilde"

**2. Projectile Motion with Horizontal Variable Acceleration?**

First, the mathematical relevance would be writing down Newton's second law in vector component form. The "assumption" you mention is quite real, namely that near the surface of the earth the force of gravity acting on us is approximately constant and points down towards the surface.Newton's second law of motion would be:-mg = ma_y0 = m*a_xwhere y direction is vertical and x horizontal, notice that mass cancels out.For your question Newton's second law would give...a_y = -ga_x = some constantorax = f(t), some function of time.Acceleration is the second derivative of position so either of these would give you a new equation to find x(t) and y(t). Now, since you are in high school physics I would ask if you've have pre-calculus or calculus 1 yet. If not then going much further may not help you see how things change but I will add a little more.What you get is a differential equation and there are techniques for solving these. You may be at the level where you have not yet learnt Newton's laws of motion but are just learning Kinematics. That's okay, we can start from that point.a_x = dv_x/dt = d/dt(dx/dt)a_y = dv_y/dt = d/dt(dy/dt)These are the equations that relate acceleration to velocity and position. The kinematics you learnt came from these equations with a_x = 0 and a_y = -g.For your first question, what if there is a constant acceleration in the x direction, this will in fact produce a parabola but the symmetry axis will not be the vertical axis but the axis defined by the acceleration vector (a_x, a_y).For a time dependent acceleration you need to "integrate" the second derivative twice and find the correct x(t), and y(t). That is a basic procedure.In the real world we have air resistance that depends on the velocity of the projectile, the force is proportional to (-v_x, -v_y). The equations of motion with "linear" air resistance are solvable and you get a correction to the parabola path that involves exponential functions. You can look this up on Wikipedia, search projectiles with air resistance.

**3. Placing text between horizontal lines generated by **

**ule: equispacing**

I have come to grips with this concept and have a solution that illustrates this pretty well:Realizing that after each new line the drawing of a line starts at the base of where a letter would be (the baseline of a glyph), you raise the line below the text. How much? user11232 uses 60% of the baseline skip. Another way would be to use the height of some upper case letter directly e. g. fontcharhtfont"004D will result in the height of an M in points. see Is baselineskip automatically defined?.If you are not happy with the optical results due to letters width depth for example, you could even make it 70% of the height of an upper case letter (just add . 7 in front of fontcharht). Draw line above and below text and see what happens. The result is not optically pleasing

**4. Horizontal brace above another brace node in TikZ**

I've never really been a fan of the remember picture and overlay trick/kludge/workaround to get these kinds of effects. So, I am stubbornly sticking with the text effects along path decoration

**5. pgfplots: Changing horizontal spacing in the legend [duplicate]**

The easiest way to do this is to specify the text width for the labels in the legend. See the second example below.I have added a text height to make all the letters sit on an even baseline. For uniformity I have also specified a text depth, even though pgfplots already sets one.

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