PFuser1232
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Homework Statement
Particles ##a## and ##b## move in opposite directions around a circle with angular speed ##ω##. At ##t = 0## they are both at the point ##\vec{r} = l \hat{\jmath}## where ##l## the radius of the circle.
Find the velocity of ##a## relative to ##b##
Homework Equations
$$\hat{\theta} = -\hat{\imath} \sin{\theta} + \hat{\jmath} \cos{\theta}$$
$$\vec{v}_{a/b} = \vec{v}_{a/O} - \vec{v}_{b/O}$$
$$\theta = \theta_0 + \dot{\theta}t$$
$$\vec{v} = \dot{\theta} r \hat{\theta}$$
The Attempt at a Solution
First, I found the tangential unit vector as a function of time for each of the particles, setting the initial angle to ##\frac{\pi}{2}## for both ##a## and ##b##. Also, according to the diagram shown in the source, ##a## is moving clockwise (towards the ##\hat{\imath}## unit vector) and ##b## is moving anticlockwise; so I set ##\dot{\theta}## equal to ##\omega## for ##b## and ##-\omega## for ##a##.
I then found the tangential velocities of both ##a## and ##b##, setting ##r## equal to ##l## for both. After doing the math and cleaning up the vector components using trigonometric identities, I got:
$$\vec{v}_{a/b} = 2l\omega \cos{\omega t} \hat{\imath}$$
I have no answer booklet though, so I don't know whether that's correct. Did I get it right, or did I go wrong somewhere?
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