How can I integrate an acceleration vector in polar coordinates?

In summary, the conversation discusses integrating an acceleration vector in polar coordinates to obtain the velocity vector. It is mentioned that integrating in Cartesian coordinates is easier because the unit vectors do not change direction, while in polar coordinates, the unit vector e_r changes direction. It is suggested to integrate in Cartesian coordinates and then transform the result if polar coordinates are needed.
  • #1
AirForceOne
49
0
Hi,

Say I have an acceleration vector in polar coordinates: a = -30e_r where the unit vector e_r points in the same direction as the Cartesian unit vector j.

How can I integrate that vector so that I have the velocity vector in polar coordinates?

I know that if I have an acceleration vector in Cartesian coordinates: a = -30j, I can integrate it with respect to time to get v = (-30t+v_0y)j + (v_0x)i.

I feel like integrating an acceleration vector in Cartesian coordinates is easier because i and j do not change as the tip of the vector moves around over time. However, with polar coordinates, e_r changes direction and yeah it gets messy.

Thanks.
 
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  • #2
Simply integrate it by the easier coordinates. That's why integration in polar coordinates are considered! If Cartesian is easier, then use it and transform the result afterwards in case you need polar coordinates.
 

What are polar coordinates and how are they different from Cartesian coordinates?

Polar coordinates are a system of coordinates used to locate a point in a two-dimensional plane. They are different from Cartesian coordinates in that they use a distance and angle from a fixed point (the origin) to describe the location of a point, rather than using x and y coordinates.

How are polar coordinates useful in scientific research?

Polar coordinates are particularly useful in fields such as physics, engineering, and astronomy. They allow for the representation and analysis of circular and rotational motion, as well as complex systems with a central point of origin.

What is the process for converting between polar and Cartesian coordinates?

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

How can polar coordinates be used to graph equations?

Polar coordinates can be used to graph equations by plotting points based on the distance from the origin and the angle from the positive x-axis. The resulting graph will have a circular or symmetric shape, depending on the equation.

What are some real-world applications of polar coordinates?

Polar coordinates have many real-world applications, including navigation and mapping, antenna design, and satellite communication. They are also used in medical imaging and geographic information systems (GIS).

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