Motion in curves - Find radial and circumferential components of V and A

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Homework Help Overview

The problem involves analyzing the motion of a comet described by a position vector that is a function of time. Participants are tasked with finding the radial and circumferential components of velocity and acceleration at a specific time.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of equations for velocity and acceleration in polar coordinates, questioning whether these can be used when the position vector is defined in Cartesian coordinates. There is an exploration of how to project the velocity and acceleration vectors into radial and circumferential components.

Discussion Status

The discussion is ongoing, with participants sharing their calculations and questioning the validity of their results. Some express confusion about the presence of both time and angle in their expressions for the radial component of velocity, indicating a need for further clarification on the mathematical operations involved.

Contextual Notes

Participants are grappling with the transition from Cartesian to polar coordinates and the implications of using time-dependent position vectors in their calculations. There is a noted uncertainty about the correct interpretation of vector multiplication in this context.

thaer_dude
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Homework Statement



At time t, a comet has the position R = (t2-1)i + 2tj

At t = 2, find the radial and circumferential components of velocity and acceleration


Homework Equations


Vr = V * Ur
Vθ = V * Uθ

ar = a * Ur
aθ = a * Uθ

Ur = cosθ i + sinθ j
Uθ = -sinθ i + cosθ j

The Attempt at a Solution



I've found

v = 2ti + 2j
a= 2i

However, am I allowed to use these equations when the position vector is a function of t and not a function of θ? I'm not very good at polar coordinates so I'm really not sure if I can apply the above equations to my problem right away. Thanks
 
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so you've found the correct equations for v & a in caretseian coords, however now you must find their projection in the radial and theta directions
 
Yeah, but say I do Vr = V * Ur

I would get

Vr = (2t i + 2 j) * (cosθ i + sinθ j)
Vr = 2tcosθ i + 2sinθ j

Is that right? It strikes me as odd that the Vr I found has both θ and t in it.
 
thaer_dude said:
Yeah, but say I do Vr = V * Ur

I would get

Vr = (2t i + 2 j) * (cosθ i + sinθ j)
Vr = 2tcosθ i + 2sinθ j

Is that right? It strikes me as odd that the Vr I found has both θ and t in it.

if that is a dot product, then it should have a scalar result, not a vector
Vr = (2t i + 2 j) * (cosθ i + sinθ j) = 2tcosθ + 2sinθ

And it should be a simple exercise to write theta in terms of t
 
Last edited:
thaer_dude said:
Yeah, but say I do Vr = V * Ur

I would get

Vr = (2t i + 2 j) * (cosθ i + sinθ j)
Vr = 2tcosθ i + 2sinθ j

Is that right? It strikes me as odd that the Vr I found has both θ and t in it.
What sort of multiplication gives: (2t i + 2 j) * (cosθ i + sinθ j) = 2t*cosθ i + 2sinθ j ?
 

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