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Polar radius and position vector: two-dimensional kinematics for high school students

Physics Asked on January 12, 2021

We consider for example this image,

enter image description here
which is a polar graph of a naval unit’s on-board instrumentation with vector radius (or polar radius) $rho$ and anomaly $theta$ or polar angle.

We know that a point $P=(x,y)$ in an orthogonal Cartesian coordinate system may be identified in a polar diagram with coordinates $Pequiv(rho,theta)$ or viceversa.

enter image description here

If we consider the trajectory $Gamma$ (the curve coloured in brown) of a target and $mathbf r=mathbf r(t)$ is your position vector is it possible to say that there is an analogy between the position vector $mathbf r$ and the polar radius $rho$? Or are the two quantities distinct because the first is a vector and the polar radius is a scalar?

2 Answers

The position vector $boldsymbol{r}(t)$ is the parametrization of the curve it rides on. But converting this to a polar form $rho(theta)$ is also a parameterization of the same curve.

For example an ellipse can be parameteized with

$$ boldsymbol{r}(t) = pmatrix{x(t) y(t)} = pmatrix{ a cos t b sin t} tag{1}$$

This position vector objeys the equation of the ellipse $$ left( tfrac{x}{a} right)^2 + left( tfrac{y}{b} right)^2 = 1$$ where $a$ is the semi-major axis, and $b$ the semi-minor axis.

Now consider the polar coordinates

$$ boldsymbol{r}(t) = pmatrix{x(t) y(t)} = pmatrix{ rho cos theta rho sin theta} $$

that yields the solution

$$ rho(theta) = frac{ a b}{sqrt{ a^2 - (a^2-b^2) cos^2 theta}} tag{2}$$

Expressions (1) and (2) are equivalent to each other since both describe the same ellipse.

Correct answer by John Alexiou on January 12, 2021

The polar radius is a scalar quantity telling you how far you are from the origin.

The position vector is a vector hence has two pieces of information, magnitude (the polar radius) and direction (the angle). Since you are in 2D, you need two coordinates to unequivocally determine your position: you need the vector.

Answered by SuperCiocia on January 12, 2021

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