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Relative velocity

  1. Mar 1, 2015 #1
    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. jcsd
  3. Mar 1, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Correct.
     
  4. Mar 1, 2015 #3
    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
     
  5. Mar 1, 2015 #4
    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}##).
     
  6. Mar 1, 2015 #5
    Nice job.

    Chet
     
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