# Polar coordinates and kinematics

• Telemachus
In summary, the student is struggling with finding the polar equation for the position, velocity, and acceleration of a rocket moving upwards with a given acceleration. They have attempted to use trigonometry to find an equation for r(t), but are unsure of how to incorporate the acceleration in both the radial and angular directions. They are seeking assistance in determining the radial and angular speeds with the given information.
Telemachus

## Homework Statement

I've got some trouble and doubts with polar coordinates. I have this exercise, with a rocket going upwards, with a given acceleration. So I need to find the polar equation for the given situation for the position, the velocity and the acceleration. How should I proceed? I mean, I know I must find an equation that describes the path for r(t) in the polar form, but I'm not sure on how the acceleration fit on this.

The distance between the origin and the rocket "D" is given by the problem data.

## Homework Equations

$$r(t)=r\vec{e_r}$$
$$v(t)=\dot r\vec{e_r}+r\dot \theta\vec{e_{\theta}}$$
$$a(t)=(\ddot r-r\dot \theta^2)\vec{e_r}+(r\ddot \theta+2 \dot r \dot \theta)\vec{e_{\theta}}$$

I think that $$r(t)$$ could be: $$r(t)=\sec \theta \vec{e_r}$$, but then I don't know how to work with the acceleration on the radial direction, or over the angle direction neither. Should I use trigonometry for this?

#### Attachments

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I've tried this way, but it doesn't looks fine really

$$a=k$$, $$s=kt$$ $$x=\displaystyle\frac{k}{2}t^2$$

$$\dot r=s \cos \theta=kt \cos \theta$$

And the angular speed:
$$\dot \theta=s \sin \theta=kt \sin \theta$$

I think that the equation for r(t) should be $$\vec{r(t)}=\sec \theta \hat{e_r}$$. But I don't know how to work with it.

$$\dot r=v(t)=\dot r\vec{e_r}+r\dot \theta\vec{e_{\theta}}$$
$$\dot r=\sec r \tan r\hat{e_r}+\sec r \dot \theta \hat{e_{\theta}}$$

I don't know how to determine the radial speed and the angular speed, considering the acceleration, and I think what I've done doesn't make any sense.

Bye there.

## What are polar coordinates?

Polar coordinates are a two-dimensional coordinate system commonly used in mathematics and physics. They are represented by a distance from the origin (the pole) and an angle from a fixed reference direction (often the positive x-axis).

## How do you convert between polar and Cartesian coordinates?

To convert from polar coordinates to Cartesian coordinates, use the equations x = r cos θ and y = r sin θ, where r is the distance from the origin and θ is the angle. To convert from Cartesian coordinates to polar coordinates, use the equations r = √(x^2 + y^2) and θ = tan^-1(y/x).

## What is the difference between polar and rectangular kinematics?

Polar kinematics involves describing the motion of an object in terms of its distance from the origin and its angle of rotation. Rectangular kinematics, on the other hand, describes motion in terms of its position along the x and y axes. Both systems use different equations and calculations to analyze motion.

## How are polar coordinates used in physics?

Polar coordinates are commonly used in physics to describe the motion of objects in circular or rotational motion. They are also used in electromagnetism, fluid mechanics, and other fields to describe the direction and magnitude of vectors.

## What are some practical applications of polar coordinates and kinematics?

Polar coordinates and kinematics are used in a variety of practical applications, including navigation systems, radar and sonar technology, and astronomy. They are also used in computer graphics to create visual effects and animations.

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