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

[thaer_dude]

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

V_r = V * U_r
V_θ = V * U_θ

a_r = a * U_r
a_θ = a * U_θ

U_r = cosθ i + sinθ j
U_θ = -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

 

[lanedance]

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

 

[thaer_dude]

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.

 

[lanedance]

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

 

[SammyS]

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 ?