An Insight into Polar Co-ordinate Velocity

[mooneyes]

Homework Statement

Derive the equations for the velocity and acceleration vectors of a particle in polar coordinates.

2. The attempt at a solution

r = xi + yj
where x = rCos$$\Theta$$, y = rSin$$\Theta$$
r = r(Cos$$\Theta$$i + Sin$$\Theta$$j)
v = $$\frac{d}{dt}$$r
v = $$\frac{d}{dt}$$[r(Cos$$\Theta$$i + Sin$$\Theta$$j)]

And I know the answer is:
v = r[dot] + r(theta[dot])
But I just don't know how to actually differentiate the expression for r with respect to time, do I treat r and theta as constants, or how does it work?

Thanks.

 

[LCKurtz]

Homework Statement

Derive the equations for the velocity and acceleration vectors of a particle in polar coordinates.

2. The attempt at a solution

r = xi + yj
where x = rCos$$\Theta$$, y = rSin$$\Theta$$
r = r(Cos$$\Theta$$i + Sin$$\Theta$$j)
v = $$\frac{d}{dt}$$r
v = $$\frac{d}{dt}$$[r(Cos$$\Theta$$i + Sin$$\Theta$$j)]

And I know the answer is:
v = r[dot] + r(theta[dot])
But I just don't know how to actually differentiate the expression for r with respect to time, do I treat r and theta as constants, or how does it work?

Thanks.

Both r and θ are functions of t. So differentiate both your rcos(θ) and your rsin(θ) in your position vector using the product and chain rules. When you are done you should be able to express your answer in terms of the basis vectors

$$e_r = \langle \cos\theta,\sin\theta\rangle,\ e_\theta=\langle -\sin\theta,\cos\theta\rangle$$