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Lets say,we draw a straight line from a point A to a point B. if a body traveling under constant acceleration travels along that line from A to B.Now, if we divide that line into segments at equal intervals of time, it can be seen that length of each segments increases with the interval count in a linear way. Now draw another line parallel to AB say CD (just ...

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My take... First, consider constant velocity $v$, then by daily intuition, the average velocity $\bar v=v$ (because its the same $v$ throughout, thus the average cannot be anything other than $v$) Now consider acceleration, and the initial velocity $v_0$, Then the final velocity $v$ must be greater than $v_0$ The average $\bar v$, which is some kind of ...

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How about this: $v(t)$ is a line. This should be an OK assumption. So it's of the form $mt+b$. If you consider the interval $t_1$ through $t_2$ and draw a horizontal line of height $\frac{1}{2}(v(t_1)+v(t_2))=m \frac{1}{2}(t_1+t_2)+b$, you'll see that exactly half of the velocity line lies above the horizontal line, and exactly half lies below. Therefore ...

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Just take the derivative of the equation of state which is the ideal gas law. $(\frac{\partial V}{\partial P})_T=-\frac{RT}{P^2}$. Now substitute into the original expression for compressibility: $\kappa_T=\frac{1}{V}(\frac{RT}{P^2})$

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We remove an overall constant for simplicity. Let us use cylindrical coordinates $(\rho,\phi,z)$, where $$\tag{1} x ~=~\rho \cos\phi, \qquad y ~=~\rho \sin\phi .$$ Also assume the standard metric $$\tag{2} ds^2~=~\mathrm{d}x\odot \mathrm{d}x +\mathrm{d}y\odot \mathrm{d}y +\mathrm{d}z\odot \mathrm{d}z ~=~\mathrm{d}\rho\odot \mathrm{d}\rho ... 0 I doubt that your expression is correct. Your original equation is of the form \begin{equation*} \partial _{\mathbf{x}}\cdot \mathbf{E(x})=\rho (\mathbf{x}) \end{equation*} where \rho (\mathbf{x}) vanishes away from the x_{3}-axis. You can write \begin{equation*} \mathbf{E(x})=\mathbf{E}_{1}\mathbf{(x})+\mathbf{E}_{2}\mathbf{(x})=\partial ... 2 The "First Equation of Motion" you define is perhaps more accurately called the "First Equation of Motion with Constant Acceleration." One would need to use Calculus to calculate the change in velocity when Acceleration is not constant, but what you call a "varying variable." Your first equation which you arrive at by Algebra:$$V_f = V_o +a \Delta t ...

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