# Relative velocity

1. Mar 1, 2015

1. The problem statement, all variables and given/known data

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 $b$

2. Relevant equations

$$\hat{\theta} = -\hat{\imath} \sin{\theta} + \hat{\jmath} \cos{\theta}$$
$$\vec{v}_{a/b} = \vec{v}_{a/O} - \vec{v}_{b/O}$$
$$\theta = \theta_0 + \dot{\theta}t$$
$$\vec{v} = \dot{\theta} r \hat{\theta}$$

3. 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?

Last edited: Mar 1, 2015
2. Mar 1, 2015

Correct.

3. Mar 1, 2015

### Staff: Mentor

Are you supposed to find their relative velocities as a function of time, or are you just supposed to find it at t = 0?

Chet

4. Mar 1, 2015

I wasn't sure, so I figured writing the velocity of $a$ relative to $b$ as a function of time would be better, since I could just set $t$ equal to zero and get the initial relative velocity ($2l\omega \hat{\imath}$).

5. Mar 1, 2015

Nice job.

Chet