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Subject:Physics (Newtonian Mechanics)

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Question:

If you would drop a feather and a bowling ball in perfect vacuum from a given height (on Earth), which one would reach the ground first?

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This is something my math teacher couldn't understand. Why do both the feather and ball reach the ground at the same time? The first way to think about it is that the only force acting on them is the gravity (there is no air resistance because it's a vacuum). The force of gravity will make objects accelerate towards the Earth with a constant acceleration of approximately 9.81 $$m/s^2$$, so they will both behave the same. My other thought is that even if the objects have different masses and the force of gravity for each of them is different, so is their inertia (how much a body "resists" at changing it's velocity). In the end, on the feather acts a smaller force but its inertia is also smaller. There is a nice video with the experiment here: https://www.youtube.com/watch?v=E43-CfukEgs

Subject:Basic Math

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Question:

Calculate the following fraction:  $$\frac{\frac{2}{3}}{\frac{3}{4}} = \; ?$$

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I learned myself this quick rule: the double fraction is equal to the product of the outer numbers over the product of inner numbers: $$\,$$ $$\frac{\frac{2}{3}}{\frac{3}{4}} = \frac{2 \times 4}{3 \times 3}= \frac{8}{9}$$ This is quicker than inverting the second fraction and then doing the multiplication, it's just one step.

Subject:Calculus

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Question:

What is the derivative of the function $$f(x) = x^x$$ ?

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Well, we know to take the derivative of $$e^{u(x)}$$ so we are going to rewrite $$f(x)$$ first. Remember that $$e^{lna}=a$$ and $${ln(a^b)}=b\,lna$$. Using these we can write: \begin{align} (x^x)' &= (e^{lnx^x})'\\ & = (e^{xlnx})' \\ & = e^{xlnx}(xlnx)' \\ & = x^x [x'lnx + x (lnx)'] \\ & = x^x ( ln x +1) \end{align} In general, when something doesn't look familiar try to rearrange it in a known form, in our case as an exponential function.

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