Sure, you can express them in any coordinate system, but the difficulty here is deciding what coordinate system. For the polar case, is it the ##\hat r## and ##\hat \theta## at the current position? At the start position? The time average of the those vectors over the same interval as the velocity is averaged over?Manish_529 said:
True, so instead consider ##r=ut, \theta=\omega t##, ##u, \omega## being constants.Manish_529 said:
Let's try it for my example in post #7. ##\vec v=u\hat r+\omega \hat\theta##, so ##\vec v_{avg}=u\frac{\Delta\hat r}{\Delta t}+\omega \frac{\Delta\hat \theta}{\Delta t}##.Manish_529 said:
Since ##\hat r, \hat\theta## are always orthogonal, ##\Delta\hat r## and ##\Delta\hat \theta## will be equal.Manish_529 said:
No. If ##\theta=\omega t## then ##\Delta\theta/\Delta t=\omega##, but ##|\Delta \hat\theta|=2\sin(\omega t/2)##.Manish_529 said:
Draw the diagram. E.g. for ##\Delta\theta=\pi/3##, if ##\hat\theta_0=\hat y## then ##\hat\theta_1=-\frac 12\sqrt 3\hat x-\frac 12\hat y##,Manish_529 said:
by the way how did we get that expression for the magnitude of the change in the unit vectorharuspex said:
Again, draw the diagram. If a triangle has two sides length 1 and the included angle is ##\theta##, what is the length of the third side?Manish_529 said:
which equals ##2\sin(\theta/2)##.Manish_529 said:
I provided the algebra for that in post #16.Manish_529 said:
You just have to decide which version of the unit vectors to use. E.g. you could choose the vectors based on current position. But with that choice, the easiest way to compute it might be to calculate it in x, y coordinates first then convert, so I don't see any value in doing it.Manish_529 said:
How about you make up an example and post an attempt?Manish_529 said:
Average velocity is defined as the total displacement divided by the total time taken for that displacement. It is a vector quantity, meaning it has both magnitude and direction.
In a Cartesian coordinate system, average velocity can be calculated using the formula: v_avg = (x_f - x_i) / (t_f - t_i), where x_f and x_i are the final and initial positions, and t_f and t_i are the final and initial times. In polar coordinates, the concept remains similar, but displacement and time need to be expressed in terms of radial and angular components.
The value of average velocity itself does not change between coordinate systems, but its representation may differ. For example, in Cartesian coordinates, it may be expressed as a vector with components along the x and y axes, whereas in polar coordinates, it may be expressed in terms of radial and angular components.
The choice of coordinate system affects how average velocity is represented and understood, particularly in multi-dimensional motion. In Cartesian coordinates, the average velocity can be easily decomposed into its components, while in non-Cartesian systems, such as polar or spherical coordinates, the interpretation may involve angular relationships and radial distances, which can complicate the analysis.
Yes, average velocity can be negative, regardless of the coordinate system used. A negative average velocity indicates that the displacement is in the opposite direction to the defined positive direction of the coordinate system. This concept holds true in both Cartesian and polar coordinates, depending on how the axes are defined.