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The formula $$ \sum_{i=1}^3 p_i q_i $$ for the dot product obviously holds for the Cartesian form of the vectors only. The proposed sum of the three products of components isn't even dimensionally correct – the radial coordinates are dimensionful while the angles are dimensionless, so they just can't be added. One can't simplify the calculation much. At ...


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\begin{equation} \mathbf{v}\left(t\right)\equiv \dfrac{d\mathbf{r}}{dt}= \dot{\mathbf{r}} \tag{01} \end{equation} We'll use one upper dot for the 1st derivative with respect to $\:t\:$, for example \begin{equation} \dot{\mathbf{r}}\equiv \dfrac{d\mathbf{r}}{dt}\;, \quad \dot{\theta}\equiv \dfrac{d\theta}{dt} \tag{02} \end{equation} Now, let a system of ...


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Consider the picture below. In Cartesian coordinates $$\hat r=\cos\theta\hat i+\sin\theta\hat j,$$ and $$\hat \theta=-\sin\theta\hat i+\cos\theta\hat j.$$ Therefore $$\frac{d\hat r}{d\theta}=\hat\theta$$


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They are not always "square." Orthogonal bases like you describe are convenient for many reasons, such as the fact that a "length" can be described in easy terms, and that there is only one way to notate any given point. There are others, such as the polar coordinate system which are different. The polar system describes 2 dimensions, one linear and one ...


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Yes. It just means that the velocity is in a direction opposite the direction of your reference frame. If you make "down = positive" then $g$ would be positive, and so would the velocity.


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Let me answer this question systematically. 1)The first is how to think of causality and locality. Locality (sometimes people use locality to talk about microcausality but that's not very important) is the statement that two events cannot communicate with each other if they are separated by spacelike distances. So what does this mean? Suppose you have a ...


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The true meaning of all the trouble is that coordinates are just tools for calculations and don't have any intrinsic meaning in GR. You should never interpret the coordinates alone. These can be very misleading and in particular become singular in some regions of spacetime (because coordinates are local). And never trust the labels of coordinates: Fact that ...


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It's the time coordinate of an event in the $S$ frame (the coordinate frame you're moving with respect to), then a fixed time. Important remark: the equation that you are using does not give the decrease (not inflation) in length of an object as measured by an observer who's moving with respect to the $S$ frame, but the $x$ coordinate transformation between ...


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What you have read is only valid in a vacuum. With air resistance the drag is a function of the total velocity, so in reality the deceleration on each axis also depends on the other.


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Let the velocity at instant be v(vector) = (Vx) i + (Vy) j. i and j denote unit vectors, along x and y axis respectively. dv/dt = a(acceleration) = - gj. dv/dt = [(Vx(final) -Vx(initial))i + (Vy(final) - Vy(initial))j]/dt = - gj. By initial and final I mean Vx and Vy at time t and t + dt. As the resulatant is only along the y axis the X component must be 0 ...


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If you take down as positive then displacement $s = +30$ m, initial velocity $v_i = +8$ ms$^{-1}$ and acceleration $a = +10$ ms$^{-2}$. Using the constant acceleration kinematic equation $s = v_i t + \frac 1 2 a t^2$ where $t$ is the time gives $$(+30) = (+8)t+\frac 1 2 (+10)t^2$$ If you take up as positive then displacement $s = -30$ m, initial velocity ...


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One way to establish that $r$ in Schwarzschild coordinates is equivalent to the spherical radial coordinate is its asymptotic behaviour, namely for $r\to\infty$ the metric tends to Minkowski, and the weak field approximation yields Newton's gravitational potential. One way to establish that the parameter $M$ is indeed the mass of the spacetime is to ...


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In this nice reference the autor assumes the relativity principle + homogeneity + isotropy and deduce the general coordinate transformations which contains both Lorentz and Galileo transformations. Further he imposes the postulate of the constancy of the speed of light, restricting the transformations to be the Lorentz type.



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